Editing Genetic drift (section)

==Mathematical models==

Mathematical models of genetic drift can be designed using either [[branching process]]es or a [[diffusion equation]] describing changes in allele frequency in an [[idealised population]].<ref>{{cite journal | vauthors = Wahl LM | title = Fixation when N and s vary: classic approaches give elegant new results | journal = Genetics | volume = 188 | issue = 4 | pages = 783–5 | date = August 2011 | pmid = 21828279 | pmc = 3176088 | doi = 10.1534/genetics.111.131748 | publisher = [[Genetics Society of America]] }}</ref>

===Wright–Fisher<!--this has c. 6x as many hits as Fisher-Wright, so please leave it alone--> model===

Consider a gene with two alleles, '''A''' or '''B'''. In [[Ploidy#Diploid|diploid]]y, populations consisting of ''N'' individuals have  2''N'' copies of each gene. An individual can have two copies of the same allele or two different alleles. The frequency of one allele is assigned ''p'' and the other ''q''. The Wright–Fisher model (named after [[Sewall Wright]] and [[Ronald Fisher]]) assumes that generations do not overlap (for example, [[annual plant]]s have exactly one generation per year) and that each copy of the gene found in the new generation is drawn independently at random from all copies of the gene in the old generation. The formula to calculate the probability of obtaining ''k'' copies of an allele that had frequency ''p'' in the last generation is then<ref name="Hartl_p112">{{harvnb|Hartl|Clark|2007|p=112}}</ref><ref>{{harvnb|Tian|2008|p=11}}</ref>

:<math>\frac{(2N)!}{k!(2N-k)!} p^k q^{2N-k} </math>

where the symbol "'''!'''" signifies the [[factorial]] function. This expression can also be formulated using the [[binomial coefficient]],

:<math>{2N \choose k}  p^k q^{2N-k} </math>


===Moran model===

The [[Moran process|Moran model]] assumes overlapping generations. At each time step, one individual is chosen to reproduce and one individual is chosen to die. So in each timestep, the number of copies of a given allele can go up by one, go down by one, or can stay the same. This means that the [[Stochastic matrix|transition matrix]] is [[tridiagonal matrix|tridiagonal]], which means that mathematical solutions are easier for the Moran model than for the Wright–Fisher model. On the other hand, [[computer simulation]]s are usually easier to perform using the Wright–Fisher model, because fewer time steps need to be calculated. In the Moran model, it takes ''N'' timesteps to get through one generation, where ''N'' is the [[effective population size]]. In the Wright–Fisher model, it takes just one.<ref>{{Cite journal | last1 = Moran | first1 = P. A. P. | author-link = Pat Moran (statistician) | doi = 10.1017/S0305004100033193 | title = Random processes in genetics | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]| volume = 54 |issue=1| pages = 60–71| year = 1958 | bibcode = 1958PCPS...54...60M| s2cid = 85823386 }}</ref>

In practice, the Moran and Wright–Fisher models give qualitatively similar results, but genetic drift runs twice as fast in the Moran model.

===Other models of drift===

If the variance in the number of offspring is much greater than that given by the binomial distribution assumed by the Wright–Fisher model, then given the same overall speed of genetic drift (the variance effective population size), genetic drift is a less powerful force compared to selection.<ref name="Charlesworth09" /> Even for the same variance, if higher [[Moment (mathematics)|moments]] of the offspring number distribution exceed those of the binomial distribution then again the force of genetic drift is substantially weakened.<ref>{{cite journal | vauthors = Der R, Epstein CL, Plotkin JB | title = Generalized population models and the nature of genetic drift | journal = Theoretical Population Biology | volume = 80 | issue = 2 | pages = 80–99 | date = September 2011 | pmid = 21718713 | doi = 10.1016/j.tpb.2011.06.004 | publisher = [[Elsevier]] | bibcode = 2011TPBio..80...80D | author-link2 = Charles Epstein (mathematician) }}</ref>

===Random effects other than sampling error===

Random changes in allele frequencies can also be caused by effects other than [[sampling error]], for example random changes in selection pressure.<ref>{{harvnb|Li|Graur|1991|p=28}}</ref>

One important alternative source of [[Stochastic#Biology|stochastic]]ity, perhaps more important than genetic drift, is [[genetic hitchhiking|genetic draft]].<ref name="gillespie 2001">{{cite journal | vauthors = Gillespie JH | title = Is the population size of a species relevant to its evolution? | journal = Evolution; International Journal of Organic Evolution | volume = 55 | issue = 11 | pages = 2161–9 | date = November 2001 | pmid = 11794777 | doi = 10.1111/j.0014-3820.2001.tb00732.x | publisher = [[John Wiley & Sons]] for the [[Society for the Study of Evolution]] | s2cid = 221735887 | author-link = John H. Gillespie | doi-access = free }}</ref> Genetic draft is the effect on a [[Locus (genetics)|locus]] by selection on [[genetic linkage|linked]] loci. The mathematical properties of genetic draft are different from those of genetic drift.<ref>{{cite journal | vauthors = Neher RA, Shraiman BI | title = Genetic draft and quasi-neutrality in large facultatively sexual populations | journal = Genetics | volume = 188 | issue = 4 | pages = 975–96 | date = August 2011 | pmid = 21625002 | pmc = 3176096 | doi = 10.1534/genetics.111.128876 | publisher = Genetics Society of America | arxiv = 1108.1635 }}</ref> The direction of the random change in allele frequency is [[autocorrelation|autocorrelated]] across generations.<ref name="Masel 2011" />