- Effective population size
In

population genetics , the concept of**effective population size**"N"_{"e"}was introduced by the Americangeneticist Sewall Wright , who wrote two landmark papers on it (Wright 1931, 1938). He defined it as "the number of breeding individuals in anidealized population that would show the same amount of dispersion of allele frequencies under randomgenetic drift or the same amount ofinbreeding as the population under consideration". It is a basic parameter in many models inpopulation genetics . The effective population size is usually smaller than the absolutepopulation size ("N"). See alsosmall population size .**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-statistics .**Variance effective size**In the Wright-Fisher idealized population model, the

conditional variance of the allele frequency $p\text{'}$, given theallele frequency $p$ in the previous generation, is:$operatorname\{var\}(p\text{'}\; mid\; p)=\; \{p(1-p)\; over\; 2N\}.$

Let $widehatoperatorname\{var\}(p\text{'}|p)$ denote the same, typically larger, variance in the actual population under consideration. The variance effective population size $N\_e^\{(v)\}$ is defined as the size of an idealized population with the same variance. This is found by equating $widehatoperatorname\{var\}(p\text{'}|p)$ with $operatorname\{var\}(p\text{'}|p)$ and solving for $N$ which gives

:$N\_e^\{(v)\}\; =\; \{p(1-p)\; over\; 2\; widehat\{operatorname\{var(p)\}.$

**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

:$F\_t\; =\; frac\{1\}\{N\}left(frac\{1+F\_\{t-2\{2\}\; ight)+left(1-frac\{1\}\{N\}\; ight)F\_\{t-1\}.$

Using Panmictic Index (1 − "F") instead of inbreeding coefficient, we get the approximate reccurrence equation

:$1-F\_t\; =\; P\_t\; =\; P\_0left(1-frac\{1\}\{2N\}\; ight)^t.$

The difference per generation is

:$frac\{P\_\{t+1\{P\_t\}\; =\; 1-frac\{1\}\{2N\}.$

The inbreeding effective size can be found by solving

:$frac\{P\_\{t+1\{P\_t\}\; =\; 1-frac\{1\}\{2N\_e^\{(F).$

This is

:$N\_e^\{(F)\}\; =\; frac\{1\}\{2left(1-frac\{P\_\{t+1\{P\_t\}\; ight)\}$

although researchers rarely use this equation directly.

**Examples****Variations in population size**Population size varies over time. Suppose there are "t" non-overlapping

generation s, then effective population size is given by theharmonic mean of the population sizes::$\{1\; over\; N\_e\}\; =\; \{1\; over\; t\}\; sum\_\{i=1\}^t\; \{1\; over\; N\_i\}$

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::

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 .Another way of calculating it

Equation for calculating Ne for populations with variation in family size: "N"

_{"e"}= (4"N")/("V"_{"k"}+ 2) Where "V"_{"k"}is the variance in population size. Large variance in family size is bad because large variations in family size lead to inbreeding.**Dioeciousness**If a population is

dioecious , i.e. there is noself-fertilisation then:$N\_e\; =\; N\; +\; egin\{matrix\}\; frac\{1\}\{2\}\; end\{matrix\}$

or more generally,

:$N\_e\; =\; N\; +\; egin\{matrix\}\; frac\{D\}\{2\}\; end\{matrix\}$

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::$N\_e\; =\; N\; +\; egin\{matrix\}\; frac\{1\}\{2\}\; approx\; N\; end\{matrix\}$

**Non-Fisherian sex-ratios**When the

sex ratio of a population varies from the Fisherian 1:1 ratio, effective population size is given by::$N\_e^\{(v)\}\; =\; N\_e^\{(F)\}\; =\; \{4\; N\_m\; N\_f\; over\; N\_m\; +\; N\_f\}$

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

gamete s to the next generation. An idealized population assumes that this follows aPoisson distribution so that thevariance of the number of gametes contributed, "k" is equal to themean number contributed, i.e. 2::$operatorname\{var\}(k)\; =\; ar\{k\}\; =\; 2.$

However, in natural populations the variance is larger than this, i.e.

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

The effective population size is then given by:

:$N\_e^\{(v)\}\; =\; \{4\; N\; -\; 2D\; over\; 2\; +\; operatorname\{var\}(k)\}$

Note that if the variance of "k" is less than 2, "N"

_{"e"}is greater than "N". Heritable variation infecundity , 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 table 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 time is calculated as

: $T\; =\; sum\_\{i=0\}^infty\; ell\_i\; v\_i\; =$ average age of a reproducing individual

Then, the inbreeding effective population size is (Felsenstein 1971)

:$N\_e^\{(F)\}\; =\; frac\{N\_0T\}\{1\; +\; sum\_iell\_\{i+1\}^2v\_\{i+1\}^2(frac\{1\}\{ell\_\{i+1-frac\{1\}\{ell\_i\})\}.$

**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 $N\_0^f$ and $N\_0^m$ for the number of newborn females and males, respectively (notice lower case "f" for females,compared to upper case "F" for inbreeding).

The inbreeding effective number is calculated from

:$frac\{1\}\{N\_e^\{(F)\; =\; frac\{1\}\{4T\}left\{frac\{1\}\{N\_0^f\}+frac\{1\}\{N\_0^m\}\; +\; sum\_ileft(ell\_\{i+1\}^f\; ight)^2left(v\_\{i+1\}^f\; ight)^2left(frac\{1\}\{ell\_\{i+1\}^f\}-frac\{1\}\{ell\_i^f\}\; ight)\; +\; sum\_ileft(ell\_\{i+1\}^m\; ight)^2left(v\_\{i+1\}^m\; ight)^2left(frac\{1\}\{ell\_\{i+1\}^m\}-frac\{1\}\{ell\_i^m\}\; ight)\; ight\}.$

**ee also***

Minimum viable population **References*** Emigh, T. H. and E. Pollak (1979). Fixation probabilities and effective population numbers in diploid populations with overlapping generations. "Theoretical Population Biology"

**15**: 86-107.

*

* Johnson, D. L. (1977). Inbreeding in populations with overlapping generations. "Genetics"**87**:581-591.

* Kempthorne, O. (1957, [1969] ). An Introduction to Genetic Statistics. 1969 Printing is Iowa State University Press.

* Wright, S. (1931). Evolution in Mendelian populations. "Genetics"**16**: 97-159 [*http://www.esp.org/foundations/genetics/classical/holdings/w/sw-31.pdf Offsite pdf file*]

* Wright, S. (1938). Size of population and breeding structure in relation to evolution. "Science"**87**:430-431**External links*** http://darwin.eeb.uconn.edu/eeb348/lecture-notes/drift/node7.html

* http://www.zoology.ubc.ca/~whitlock/bio434/LectureNotes/05.EffectiveSize/EffectiveSize.html

* http://www.kursus.kvl.dk/shares/vetgen/_Popgen/genetics/3/6.htm

* http://wiki.cotch.net/index.php/Effective_population_size

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