Difference between revisions of "Genetic Drift"
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<math>N</math> is the (diploid) population size. | <math>N</math> is the (diploid) population size. | ||
| − | The derivation of this stems from the probability of two allele copies being identical by descent over a period of time. There are 2''N'' total allele copies in the population. The probability of two alleles arising from the exact same copy in the previous generation is <math>1/2N</math>. The probability that this did ''not'' occur is <math>1 - 1/2N</math> per generation. There is an independent chance of this each ''t'' generation so these probabilities are multiplied: <math>(1 - 1/2N)^t</math>. (If two randomly selected allele copies arose from the same copy in the previous generation then they cannot be heterozygous; thus, this is a way to keep track of the change in average heterozygosity in the population.) Wright (1922) formulated this as the rate of increase of identity-by-descent in terms of a "fixation" index ''F'' (variation is lost when a single allele reaches fxation, 100%, in the population). The increase in homozygosity (fixation) is one minus the decrease in heterozygosity. <math>F_t = 1-H_t</math>. So, <math>F_t = F_0 1 - (1-\frac{1}{2N})^t</math>. | + | The derivation of this stems from the probability of two allele copies being identical by descent over a period of time. There are 2''N'' total allele copies in the population. The probability of two alleles arising from the exact same copy in the previous generation is <math>1/2N</math>. The probability that this did ''not'' occur is <math>1 - 1/2N</math> per generation. There is an independent chance of this each ''t'' generation so these probabilities are multiplied: <math>(1 - 1/2N)^t</math>. (If two randomly selected allele copies arose from the same copy in the previous generation then they cannot be heterozygous; thus, this is a way to keep track of the change in average heterozygosity in the population.) Wright (1922)<ref>. Wright, S. (1922) Coeffcients of inbreeding and relationship. American Naturalist 56: 330-338.[https://scholar.google.com/scholar?cluster=17435973857108896516]</ref> formulated this as the rate of increase of identity-by-descent in terms of a "fixation" index ''F'' (variation is lost when a single allele reaches fxation, 100%, in the population). The increase in homozygosity (fixation) is one minus the decrease in heterozygosity. <math>F_t = 1-H_t</math>. So, <math>F_t = F_0 1 - (1-\frac{1}{2N})^t</math>. |
This can be simplified by using an exponential function approximation. | This can be simplified by using an exponential function approximation. | ||
Revision as of 17:27, 23 January 2016
Alleles are sampled from a population to make up the next generation. Anytime a random sample is taken it is likely to deviate slightly from the original population. Genetic drift is evolutionary sampling error. These deviations are not biased in any single direction towards an increase or decrease in frequency of a particular allele. However, these small deviations will accumulate over several generations and ultimately alleles will be lost until only a single allele remains in the population (assuming it is not reintroduced by mutation or from another population).
Exact Binomial Probabilities
Average Loss of Heterozygosity
If a small population becomes isolated so that there is no introduction of new alleles (for example in a captive breeding program or as a small number of founders on a new island) and genetic drift is a major force, the average loss of heterozygosity in the genome over multiple generations can be estimated.
[math]H_t = H_0 (1-\frac{1}{2N})^t[/math]
[math]H_t[/math] is the remaining heterozygosity at generation [math]t[/math].
[math]H_0[/math] is the initial heterozygosity at generation zero.
[math]t[/math] is the time in generations.
[math]N[/math] is the (diploid) population size.
The derivation of this stems from the probability of two allele copies being identical by descent over a period of time. There are 2N total allele copies in the population. The probability of two alleles arising from the exact same copy in the previous generation is [math]1/2N[/math]. The probability that this did not occur is [math]1 - 1/2N[/math] per generation. There is an independent chance of this each t generation so these probabilities are multiplied: [math](1 - 1/2N)^t[/math]. (If two randomly selected allele copies arose from the same copy in the previous generation then they cannot be heterozygous; thus, this is a way to keep track of the change in average heterozygosity in the population.) Wright (1922)[1] formulated this as the rate of increase of identity-by-descent in terms of a "fixation" index F (variation is lost when a single allele reaches fxation, 100%, in the population). The increase in homozygosity (fixation) is one minus the decrease in heterozygosity. [math]F_t = 1-H_t[/math]. So, [math]F_t = F_0 1 - (1-\frac{1}{2N})^t[/math].
This can be simplified by using an exponential function approximation.
[math](1-\frac{1}{2N})^t \approx e^{- t/ 2N}[/math]
[math]H_t \approx H_0 e^{- t/ 2N}[/math]
The time until only half of the original genetic variation remains can be directly estimated as a rule-of-thumb.
[math]\frac{1}{2} = \frac{H_t}{H_0} \approx e^{- t/ 2N}[/math]
[math]\log{\frac{1}{2}} \approx \log{e^{- t/ 2N}}[/math]
[math]-0.6931 \approx - t/ 2N[/math]
[math]1.386 N \approx t [/math]
So, on average half of the genetic variation will be lost due to drift over a period of 1.386 N generations, where N is the diploid population size.
As an example, the 'alalā (Hawaiian crow, Corvus hawaiiensis) became extinct in the wild in 2002 and the species only exists as about 100 individuals maintained in captivity. This predicts that it will take 138 generations for the genetic variation (heterozygosity) of the species to decline by half (if the number of captive individuals remains constant each generation).
In another example, the resident human population of Pitcairn Island is about 50 individuals. If they became completely isolated from the rest of the world and maintained the same population size the majority of their current genetic variation is predicted to still be present in the population 2,000 years from now (assuming a generation time of 30 years).
Of course these examples assume an ideal random-mating population of constant size where each individual has an equal chance of reproducing. There are many ways this scenario can be altered to increase or decrease the rate of loss of genetic variation.
