Editing Effective population size

{{short description|Ecological concept}}
{{Technical|date=November 2020}}
The '''effective population size''' (''N''<sub>''e''</sub>) is the size of an [[idealised population]] that would experience the same rate of [[genetic drift]] as the real population.<ref>{{cite web|url=http://www.blackwellpublishing.com/ridley/a-z/Effective_population_size.asp|title=Effective population size|access-date=4 March 2018|work=[[Blackwell Publishing]]}}</ref> Idealised populations are those following simple one-[[Locus (genetics)|locus]] models that comply with assumptions of the [[neutral theory of molecular evolution]]. The effective population size is normally smaller than the [[population size|census population size]] ''N'', partly because chance events prevent some individuals from breeding, and partly due to [[background selection]] and [[genetic hitchhiking]].

The same real population could have a different effective population size for different properties of interest, such as genetic drift (or more precisely, the speed of [[Coalescent theory#Time to coalescence|coalescence]]) over one generation vs. over many generations. Within a species, [[Locus (genetics)|areas]] of the genome that have more [[gene]]s and/or less [[genetic recombination]] tend to have lower effective population sizes, because of the effects of selection at linked sites. In a population with selection at many loci and abundant [[linkage disequilibrium]], 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]] in 1931 by the [[United States|American]] [[geneticist]] [[Sewall Wright]].<ref>{{cite journal |author=Wright S |year=1931 |title=Evolution in Mendelian populations |journal=[[Genetics (journal)|Genetics]] |volume=16 |issue=2 |pages=97–159 |doi=10.1093/genetics/16.2.97 |pmid=17246615 |pmc=1201091 |url=http://www.esp.org/foundations/genetics/classical/holdings/w/sw-31.pdf}}</ref><ref>{{cite journal |author=Wright S |year=1938 |title=Size of population and breeding structure in relation to evolution |journal=[[Science (journal)|Science]] |volume=87 |pages=430–431 | doi = 10.1126/science.87.2263.425-a |issue=2263}}</ref> Some versions of the effective population size are used in wildlife conservation.  

== Empirical measurements ==

In a rare experiment that directly measured genetic drift one generation at a time, in ''Drosophila'' populations of census size 16, the effective population size was 11.5.<ref>{{cite journal|last=Buri|first=P|journal=Evolution|year=1956|volume=10|issue=4|pages=367–402|title=Gene frequency in small populations of mutant Drosophila|doi=10.2307/2406998|jstor=2406998}}</ref> This measurement was achieved through studying changes in the frequency of a neutral allele from one generation to another in over 100 replicate populations.

More commonly, effective population size is estimated indirectly by comparing data on current within-species [[nucleotide diversity|genetic diversity]] to theoretical expectations. According to the [[neutral theory of molecular evolution]], an idealised diploid population will have a pairwise [[nucleotide diversity]] equal to 4<math>\mu</math>''N''<sub>''e''</sub>, where <math>\mu</math> is the [[mutation rate]]. The effective population size can therefore be estimated empirically by dividing the nucleotide diversity by 4<math>\mu</math>.<ref name="Lynch 2003" /> This captures the cumulative effects of genetic drift, genetic hitchhiking, and background selection over longer timescales. More advanced methods, permitting a changing effective population size over time, have also been developed.<ref name="Weinreich 2023">{{cite book |last1=Weinreich |first1=Daniel M. |title=The foundations of population genetics |date=2023 |publisher=The MIT Press |location=Cambridge, Massachusetts |isbn=978-0262047579}}</ref>

The effective size measured to reflect these longer timescales may have little relationship to the number of individuals physically present in a population.<ref>
{{cite journal
|title=Is the population size of a species relevant to its evolution?
|journal=Evolution
|year=2001
|volume=55
|pages=2161–2169
|doi=10.1111/j.0014-3820.2001.tb00732.x 
|author=Gillespie, JH
|pmid=11794777
|issue=11|doi-access=free
}}
</ref> Measured 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.<ref>{{cite journal |title=Toward a selection theory of molecular evolution|journal=Evolution|year=2008|volume=62|pages=255–265|doi=10.1111/j.1558-5646.2007.00308.x |author=Hahn, Matthew W.
|pmid=18302709|issue=2|doi-access=free}}</ref><ref>{{cite journal|title=Rethinking Hardy–Weinberg and genetic drift in undergraduate biology|journal=BioEssays|year=2012|doi=10.1002/bies.201100178|author=Masel, Joanna|author-link=Joanna Masel|pmid=22576789 |volume=34|issue=8|pages=701–10|s2cid=28513167}}</ref> Sojourn times are proportional to N in neutral theory, but for alleles under selection, sojourn times are proportional to log(N). [[Genetic hitchhiking]] 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.<ref>{{cite journal |last1=Neher |first1=Richard A. |title=Genetic Draft, Selective Interference, and Population Genetics of Rapid Adaptation |journal=Annual Review of Ecology, Evolution, and Systematics |date=23 November 2013 |volume=44 |issue=1 |pages=195–215 |doi=10.1146/annurev-ecolsys-110512-135920|arxiv=1302.1148 }}</ref>

If the [[Genetic linkage#Linkage map|recombination map]] of [[Genetic linkage#Recombination frequency|recombination frequencies]] along [[chromosome]]s is known, ''N''<sub>''e''</sub> can be inferred from ''r''<sub>P</sub><sup>2</sup> = 1 / (1+4''N''<sub>''e''</sub> ''r''), where ''r''<sub>P</sub> is the [[Pearson correlation coefficient]] between loci.<ref>{{cite journal |last1=Tenesa |first1=Albert |last2=Navarro |first2=Pau |last3=Hayes |first3=Ben J. |last4=Duffy |first4=David L. |last5=Clarke |first5=Geraldine M. |last6=Goddard |first6=Mike E. |last7=Visscher |first7=Peter M. |title=Recent human effective population size estimated from linkage disequilibrium |journal=Genome Research |date=April 2007 |volume=17 |issue=4 |pages=520–526 |doi=10.1101/gr.6023607|pmid=17351134 |pmc=1832099 |hdl=20.500.11820/b0ffcebe-9ce4-4efe-8bd9-70327945df8b |hdl-access=free }}</ref> This expression can be interpreted as the probability that two [[Lineage (genetic)|lineages]] coalesce before one allele on either lineage recombines onto some third lineage.<ref name="Weinreich 2023"></ref>

A survey of publications on 102 mostly wildlife animal and plant species yielded 192 ''N''<sub>''e''</sub>/''N'' ratios. Seven different estimation methods were used in the surveyed studies. Accordingly, the ratios ranged widely from 10<sup>''-6''</sup> for Pacific oysters to 0.994 for humans, with an average of 0.34 across the examined species. Based on these data they subsequently estimated more comprehensive ratios, accounting for fluctuations in population size, variance in family size and unequal sex-ratio. These ratios average to only 0.10-0.11.<ref name="Frankham 1995">{{Cite journal| volume = 66| pages = 95–107|author1=R. Frankham | title = Effective population size/adult population size ratios in wildlife: a review| journal = Genetics Research | year = 1995| doi = 10.1017/S0016672300034455 | issue = 2| doi-access = free}}</ref>

A genealogical analysis of human hunter-gatherers ([[Eskimo]]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.<ref name="Matsumura 2008">{{Cite journal| volume = 275| pages = 1501–1508|author1=S. Matsumura|author2=P. Forster| title = Generation time and effective population size in Polar Eskimos.| journal = Proc Biol Sci| year = 2008| doi = 10.1098/rspb.2007.1724| issue = 1642| pmid = 18364314| pmc = 2602656}}</ref>

== 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]] 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.<ref name="Neher 2011" /><ref>{{cite journal
|title=Limits to the Rate of Adaptive Substitution in Sexual Populations|journal=PLOS Genetics|year=2012|volume=8|issue=6|pages=e1002740|doi=10.1371/journal.pgen.1002740|author1=Daniel B. Weissman |author2=Nicholas H. Barton |pmid=22685419 |pmc=3369949 |doi-access=free }}</ref> 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.<ref>{{cite book|last=Lynch|first=Michael|title=The Origins of Genome Architecture |year=2007|publisher=Sinauer Associates|isbn=978-0-87893-484-3}}</ref><ref>{{cite journal|author1=Rajon, E. |author2=Masel, J. |author2-link=Joanna Masel | title = Evolution of molecular error rates and the consequences for evolvability| journal = PNAS| year=2011 |volume = 108| issue = 3| pages = 1082–1087 |doi=10.1073/pnas.1012918108| pmid = 21199946| pmc = 3024668 |bibcode=2011PNAS..108.1082R |doi-access=free }}</ref>

== History of theory ==
[[Ronald Fisher]] and [[Sewall Wright]] originally defined effective population size as "the number of breeding individuals in an [[idealised population]] that would show the same amount of dispersion of [[allele frequency|allele frequencies]] under random [[genetic drift]] or the same amount of [[inbreeding]] as the population under consideration". This implied two potentially different effective population sizes, based either on the one-generation increase in variance across replicate populations '''(variance effective population size)''', or on the one-generation change in the inbreeding coefficient '''(inbreeding effective population size)'''. These two are closely linked, and derived from [[F-statistics]], but they are not identical.<ref>{{cite journal |title=Wright and Fisher on Inbreeding and Random Drift |journal=Genetics |author=James F. Crow |author-link=James F. Crow |year=2010 |volume=184 |issue=3 |pages=609–611 |doi=10.1534/genetics.109.110023 |pmc=2845331 |pmid=20332416}}</ref>

Today, the effective population size is usually estimated empirically with respect to the amount of within-species [[nucleotide diversity|genetic diversity]] divided by the [[mutation rate]], yielding a '''coalescent effective population size''' that reflects the cumulative effects of genetic drift, background selection, and genetic hitchhiking over longer time periods.<ref name="Lynch 2003">{{cite journal |author=Lynch, M. |author2=Conery, J.S. |title=The origins of genome complexity |journal=Science|year=2003|volume=302|issue=5649 |pages=1401–1404 |doi=10.1126/science.1089370 |pmid=14631042|bibcode=2003Sci...302.1401L |citeseerx=10.1.1.135.974 |s2cid=11246091 }}</ref> Another important effective population size is the '''selection effective population size''' 1/s<sub>critical</sub>, where s<sub>critical</sub> is the critical value of the [[selection coefficient]] at which selection becomes more important than [[genetic drift]].<ref name="Neher 2011">{{Cite journal| volume = 188| pages = 975–996|author1=R.A. Neher |author2=B.I. Shraiman | title = Genetic Draft and Quasi-Neutrality in Large Facultatively Sexual Populations| journal = Genetics| year = 2011| doi = 10.1534/genetics.111.128876| issue = 4| pmid = 21625002| pmc = 3176096| arxiv = 1108.1635}}</ref>

=== Variance effective size ===
In the [[Idealized population|Wright-Fisher idealized population model]], the [[conditional variance]] of the allele frequency <math>p'</math>, given the [[allele frequency]] <math>p</math> in the previous generation, is

:<math>\operatorname{var}(p' \mid p)= {p(1-p) \over 2N}.</math>

Let <math>\widehat{\operatorname{var}}(p'\mid p)</math> denote the same, typically larger, variance in the actual population under consideration.  The variance effective population size <math>N_e^{(v)}</math> is defined as the size of an idealized population with the same variance.  This is found by substituting <math>\widehat{\operatorname{var}}(p'\mid p)</math> for <math>\operatorname{var}(p'\mid p)</math> and solving for <math>N</math> which gives

:<math>N_e^{(v)} = {p(1-p) \over 2 \widehat{\operatorname{var}}(p)}.</math>

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]]s, then effective population size is given by the [[harmonic mean]] of the population sizes:<ref>{{Cite journal|last=Karlin|first=Samuel|date=1968-09-01|title=Rates of Approach to Homozygosity for Finite Stochastic Models with Variable Population Size|journal=The American Naturalist|volume=102|issue=927|pages=443–455|doi=10.1086/282557|bibcode=1968ANat..102..443K |s2cid=83824294|issn=0003-0147}}</ref>

:<math>{1 \over N_e} = {1 \over t} \sum_{i=1}^t {1 \over N_i}</math>

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:

:{|
|-
|<math>{1 \over N_e}</math>
|<math>= {\begin{matrix} \frac{1}{10} \end{matrix} + \begin{matrix} \frac{1}{100} \end{matrix} + \begin{matrix} \frac{1}{50} \end{matrix} + \begin{matrix} \frac{1}{80} \end{matrix} + \begin{matrix} \frac{1}{20} \end{matrix} + \begin{matrix} \frac{1}{500} \end{matrix} \over 6} </math>
|-
|
|<math>= {0.1945 \over 6}</math>
|-
|
|<math>= 0.032416667</math>
|-
|<math>N_e</math>
|<math>= 30.8</math>
|}

Note this is less than the [[arithmetic mean]] of the population size, which in this example is 126.7. The harmonic mean tends to be dominated by the smallest [[population bottleneck|bottleneck]] that the population goes through.

==== Dioeciousness ====

If a population is [[dioecious]], i.e. there is no [[self-fertilisation]] then

:<math>N_e = N + \begin{matrix} \frac{1}{2} \end{matrix}</math>

or more generally,

:<math>N_e = N + \begin{matrix} \frac{D}{2} \end{matrix}</math>

where ''D'' represents dioeciousness and may take the value 0 (for not dioecious) or 1 for dioecious.

When ''N'' is large, ''N''<sub>''e''</sub> approximately equals ''N'', so this is usually trivial and often ignored:

:<math>N_e = N + \begin{matrix} \frac{1}{2} \approx N \end{matrix}</math>

==== Variance in reproductive success ====

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 a [[Poisson distribution]] so that the [[variance]] of the number of gametes contributed, ''k'' is equal to the [[mean]] number contributed, i.e. 2:

:<math>\operatorname{var}(k) = \bar{k} = 2.</math>

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

:<math>\operatorname{var}(k) > 2.</math>

The effective population size is then smaller, and given by:

:<math>N_e^{(v)} = {4 N - 2D \over 2 + \operatorname{var}(k)}</math>

Note that if the variance of ''k'' is less than 2,  ''N''<sub>''e''</sub> 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''<sub>''k''</sub> = 0 and ''N''<sub>''e''</sub> = 2''N''.

==== Non-Fisherian sex-ratios ====

When the [[sex ratio]] of a population varies from the [[Ronald Fisher|Fisherian]] 1:1 ratio,  effective population size is given by:

:<math>N_e^{(v)} = N_e^{(F)} = {4 N_m N_f \over N_m + N_f}</math>

Where ''N''<sub>''m''</sub> is the number of males and ''N''<sub>''f''</sub> the number of females.  For example, with 80 males and 20 females (an absolute population size of 100):
:{|
|-
|<math>N_e</math>
|<math>= {4 \times 80 \times 20 \over 80 + 20}</math>
|-
|
|<math>={6400 \over 100}</math>
|-
|
|<math>= 64</math>
|}

Again, this results in ''N''<sub>''e''</sub> being less than ''N''.

===Inbreeding effective size===

Alternatively, the effective population size may be defined by noting how the average [[inbreeding coefficient]] changes from one generation to the next, and then defining ''N''<sub>''e''</sub> 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).<ref>{{cite book |author=Kempthorne O |year=1957 |title=An Introduction to Genetic Statistics |publisher=Iowa State University Press}}</ref>

For the idealized population, the inbreeding coefficients follow the recurrence equation

:<math>F_t = \frac{1}{N}\left(\frac{1+F_{t-2}}{2}\right)+\left(1-\frac{1}{N}\right)F_{t-1}.</math>

Using Panmictic Index (1&nbsp;&minus;&nbsp;''F'') instead of inbreeding coefficient, we get the approximate recurrence equation

:<math>1-F_t = P_t = P_0\left(1-\frac{1}{2N}\right)^t. </math>

The difference per generation is

:<math>\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N}. </math>

The inbreeding effective size can be found by solving

:<math>\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N_e^{(F)}}. </math>

This is

:<math>N_e^{(F)} = \frac{1}{2\left(1-\frac{P_{t+1}}{P_t}\right)} </math>.

==== Theory of 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:

: <math>v_i = </math> [[Fisher's reproductive value]] for age <math>i</math>,

: <math>\ell_i = </math> The chance an individual will survive to age <math>i</math>, and

: <math>N_0 = </math> The number of newborn individuals per breeding season.

The [[generation time]] is calculated as

: <math>T = \sum_{i=0}^\infty \ell_i v_i = </math> average age of a reproducing individual

Then, the inbreeding effective population size is<ref>{{cite journal |author=Felsenstein J |year=1971 |title=Inbreeding and variance effective numbers in populations with overlapping generations | journal= [[Genetics (journal)|Genetics]]|volume= 68|issue=4 |pages=581–597|doi=10.1093/genetics/68.4.581 |pmid=5166069 |pmc=1212678 }}</ref>

:<math>N_e^{(F)} = \frac{N_0T}{1 + \sum_i\ell_{i+1}^2v_{i+1}^2(\frac{1}{\ell_{i+1}}-\frac{1}{\ell_i})}.</math>

===== Diploid =====
Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure.  This was first given by Johnson,<ref>{{cite journal |author=Johnson DL |year=1977 |title=Inbreeding in populations with overlapping generations |journal=[[Genetics (journal)|Genetics]] |volume=87 |issue=3 |pages=581–591|doi=10.1093/genetics/87.3.581 |pmid=17248780 |pmc=1213763 }}</ref> but the notation more closely resembles Emigh and Pollak.<ref>{{cite journal |doi=10.1016/0040-5809(79)90028-5 |vauthors=Emigh TH, Pollak E |year=1979 |title=Fixation probabilities and effective population numbers in diploid populations with overlapping generations |journal=Theoretical Population Biology |volume=15 |issue=1 |pages=86–107|bibcode=1979TPBio..15...86E }}</ref>

Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as ''N''<sub>0</sub><sup>''ƒ''</sup> and ''N''<sub>0</sub><sup>''m''</sup> 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

:<math>
\begin{align}
\frac{1}{N_e^{(F)}} = \frac{1}{4T}\left\{\frac{1}{N_0^f}+\frac{1}{N_0^m} + \sum_i\left(\ell_{i+1}^f\right)^2\left(v_{i+1}^f\right)^2\left(\frac{1}{\ell_{i+1}^f}-\frac{1}{\ell_i^f}\right)\right. \,\,\,\,\,\,\,\, & \\
 \left. {} + \sum_i\left(\ell_{i+1}^m\right)^2\left(v_{i+1}^m\right)^2\left(\frac{1}{\ell_{i+1}^m}-\frac{1}{\ell_i^m}\right) \right\}. &
\end{align}
</math>


==See also==
* [[Minimum viable population]]
* [[Small population size]]

== References ==

{{Reflist|colwidth=30em}}

== External links ==
* {{cite web |last=Holsinger |first=Kent |title=Effective Population Size |date=2008-08-26 |publisher=University of Connecticut |url=http://darwin.eeb.uconn.edu/eeb348/lecture-notes/drift/node7.html |url-status=dead |archive-url=https://web.archive.org/web/20050524233613/http://darwin.eeb.uconn.edu/eeb348/lecture-notes/drift/node7.html |archive-date=2005-05-24 }}
* {{cite web |last=Whitlock |first=Michael |title=The Effective Population Size |year=2008 |url=http://www.zoology.ubc.ca/~whitlock/bio434/LectureNotes/05.EffectiveSize/EffectiveSize.html |work=Biology 434: Population Genetics |publisher=The University of British Columbia |access-date=2005-02-25 |archive-url=https://web.archive.org/web/20090723072549/http://www.zoology.ubc.ca/~whitlock/bio434/LectureNotes/05.EffectiveSize/EffectiveSize.html |archive-date=2009-07-23 |url-status=dead }}
* https://web.archive.org/web/20050524144622/http://www.kursus.kvl.dk/shares/vetgen/_Popgen/genetics/3/6.htm &mdash; on Københavns Universitet.

{{qg}}
{{Population genetics}}
{{modelling ecosystems|expanded=none}}

[[Category:Population genetics]]
[[Category:Population ecology]]
[[Category:Ecological metrics]]
[[Category:Quantitative genetics]]