Editing Genetic drift (section)

==Drift and fixation==
The [[Hardy–Weinberg principle]] states that within sufficiently large populations, the allele frequencies remain constant from one generation to the next unless the equilibrium is disturbed by [[gene flow|migration]], genetic [[mutation]]s, or [[Selection (biology)|selection]].<ref>{{harvnb|Ewens|2004}}</ref>
[[File:Effects of Genetic Drift on Large and Small Populations.jpg|alt= In the smaller population of frogs (N=10), there is a large effect of genetic drift, which reduces the genetic diversity in the smaller population. However, in the larger population of frogs (N=50), there is a small effect of genetic drift, which doesn't heavily affect the larger population.|thumb|399x399px|The Effects of Genetic Drift on Different Population Sizes]]
However, in finite populations, no new alleles are gained from the random sampling of alleles passed to the next generation, but the sampling can cause an existing allele to disappear. Because [[random sample|random sampling]] can remove, but not replace, an allele, and because random declines or increases in allele frequency influence expected allele distributions for the next generation, genetic drift drives a population towards genetic uniformity over time. When an allele reaches a frequency of 1 (100%) it is said to be "fixed" in the population and when an allele reaches a frequency of 0 (0%) it is lost. Smaller populations achieve fixation faster, whereas in the limit of an infinite population, fixation is not achieved. Once an allele becomes fixed, genetic drift comes to a halt, and the allele frequency cannot change unless a new allele is introduced in the population via mutation or [[gene flow]]. Thus even while genetic drift is a random, directionless process, it acts to eliminate [[genetic variation]] over time.<ref>{{harvnb|Li|Graur|1991|p=29}}</ref>

===Rate of allele frequency change due to drift===
[[File:Random genetic drift chart.png|thumb|250px|Ten simulations of random genetic drift of a single given allele with an initial frequency distribution 0.5 measured over the course of 50 generations, repeated in three reproductively synchronous populations of different sizes. In these simulations, alleles drift to loss or fixation (frequency of 0.0 or 1.0) only in the smallest population.]]
Assuming genetic drift is the only evolutionary force acting on an allele, after ''t'' generations in many replicated populations, starting with allele frequencies of ''p'' and ''q'', the variance in allele frequency across those populations is

: <math>
V_t \approx pq\left(1-\exp\left(-\frac{t}{2N_e} \right)\right)
</math><ref>{{harvnb|Barton|Briggs|Eisen|Goldstein|2007|p=417}}</ref>

===Time to fixation or loss===
Assuming genetic drift is the only evolutionary force acting on an allele, at any given time the probability that an allele will eventually become fixed in the population is simply its frequency in the population at that time.<ref>{{harvnb|Futuyma|1998|p=300}}</ref> For example, if the frequency ''p'' for allele '''A''' is 75% and the frequency ''q'' for allele '''B''' is 25%, then given unlimited time the probability '''A''' will ultimately become fixed in the population is 75% and the probability that '''B''' will become fixed is 25%.

The expected number of generations for fixation to occur is [[Proportionality (mathematics)|proportional]] to the population size, such that fixation is predicted to occur much more rapidly in smaller populations.<ref>{{cite journal | vauthors = Otto SP, Whitlock MC | title = The probability of fixation in populations of changing size | journal = Genetics | volume = 146 | issue = 2 | pages = 723–33 | date = June 1997 | pmid = 9178020 | pmc = 1208011 | url = http://www.genetics.org/content/146/2/723.full.pdf | publisher = Genetics Society of America| doi = 10.1093/genetics/146.2.723 | url-status = live | archive-date = 19 March 2015 | archive-url = https://web.archive.org/web/20150319042554/http://www.genetics.org/content/146/2/723.full.pdf | author-link1 = Sarah Otto }}</ref> Normally the effective population size, which is smaller than the total population, is used to determine these probabilities. The effective population (''N''<sub>''e''</sub>) takes into account factors such as the level of [[inbreeding]], the stage of the lifecycle in which the population is the smallest, and the fact that some neutral genes are genetically linked to others that are under selection.<ref name="Charlesworth09">{{cite journal | vauthors = Charlesworth B | title = Fundamental concepts in genetics: effective population size and patterns of molecular evolution and variation | journal = Nature Reviews. Genetics | volume = 10 | issue = 3 | pages = 195–205 | date = March 2009 | pmid = 19204717 | doi = 10.1038/nrg2526 | publisher = [[Nature Publishing Group]] | s2cid = 205484393 | author-link = Brian Charlesworth }}</ref> The effective population size may not be the same for every gene in the same population.<ref>{{cite journal | vauthors = Cutter AD, Choi JY | title = Natural selection shapes nucleotide polymorphism across the genome of the nematode Caenorhabditis briggsae | journal = Genome Research | volume = 20 | issue = 8 | pages = 1103–11 | date = August 2010 | pmid = 20508143 | pmc = 2909573 | doi = 10.1101/gr.104331.109 | publisher = [[Cold Spring Harbor Laboratory Press]] }}</ref>

One forward-looking formula used for approximating the expected time before a neutral allele becomes fixed through genetic drift, according to the Wright–Fisher model, is

: <math>
\bar{T}_\text{fixed} = \frac{-4N_e(1-p) \ln (1-p)}{p}
</math>

where ''T'' is the number of generations, ''N''<sub>''e''</sub> is the effective population size, and ''p'' is the initial frequency for the given allele. The result is the number of generations [[Expected value|expected]] to pass before fixation occurs for a given allele in a population with given size (''N''<sub>''e''</sub>) and allele frequency (''p'').<ref>{{harvnb|Hedrick|2005|p=315}}</ref>

The expected time for the neutral allele to be lost through genetic drift can be calculated as<ref name="Hartl_p112" />

: <math>
\bar{T}_\text{lost} = \frac{-4N_ep}{1-p} \ln p.
</math>

When a mutation appears only once in a population large enough for the initial frequency to be negligible, the formulas can be simplified to<ref>{{harvnb|Li|Graur|1991|p=33}}</ref>

: <math>
\bar{T}_\text{fixed} = 4N_e
</math>

for average number of generations expected before fixation of a neutral mutation, and

: <math>
\bar{T}_\text{lost} = 2 \left ( \frac{N_e}{N} \right ) \ln (2N)
</math>

for the average number of generations expected before the loss of a neutral mutation in a population of actual size N.<ref>{{harvnb|Kimura|Ohta|1971}}</ref>

===Time to loss with both drift and mutation===
The formulae above apply to an allele that is already present in a population, and which is subject to neither mutation nor natural selection. If an allele is lost by mutation much more often than it is gained by mutation, then mutation, as well as drift, may influence the time to loss. If the allele prone to mutational loss begins as fixed in the population, and is lost by mutation at rate m per replication, then the expected time in generations until its loss in a haploid population is given by

: <math>
\bar{T}_\text{lost} \approx \begin{cases}
\dfrac 1 m, \text{ if } mN_e \ll 1\\[8pt]
\dfrac{\ln{(mN_e)}+\gamma} m \text{ if } mN_e \gg 1
\end{cases}
</math>

where <math>\gamma</math> is [[Euler–Mascheroni constant|Euler's constant]].<ref>{{cite journal | vauthors = Masel J, King OD, Maughan H | title = The loss of adaptive plasticity during long periods of environmental stasis | journal = The American Naturalist | volume = 169 | issue = 1 | pages = 38–46 | date = January 2007 | pmid = 17206583 | pmc = 1766558 | doi = 10.1086/510212 | publisher = [[University of Chicago Press]] on behalf of the [[American Society of Naturalists]] | bibcode = 2007ANat..169...38M | author1-link = Joanna Masel }}</ref> The first approximation represents the waiting time until the first mutant destined for loss, with loss then occurring relatively rapidly by genetic drift, taking time {{nowrap|{{sfrac|1|''m''}} ≫ ''N''<sub>''e''</sub>.}} The second approximation represents the time needed for deterministic loss by mutation accumulation. In both cases, the time to fixation is dominated by mutation via the term {{sfrac|1|''m''}}, and is less affected by the [[effective population size]].