Fixation (population genetics)

For other uses, see Fixation.

In population genetics, fixation is the change in a gene pool from a situation where there exists at least two variants of a particular gene (allele) to a situation where only one of the alleles remains.[1] In the absence of mutation, any allele must eventually be lost completely from the population or fixed (permanently established in the population).[2] Whether a gene will ultimately begin lost of fixed is dependent on selection coefficients and chance fluctuations in allelic proportions.[3] Fixation can refer to a gene in general or particular nucleotide position in the DNA chain (locus).

In the process of substitution, a previously non-existent allele arises by mutation and undergoes fixation by spreading through the population by random genetic drift and/or positive selection. Once the frequency of the allele is at 100%, i.e. being the only gene variant present in any member, it is said to be "fixed" in the population.[1]

Similarly, genetic differences between taxa are said to have been fixed in each species.

Probability of fixation

Under conditions of genetic drift alone, every finite set of genes or alleles has a "coalescent point" at which all descendants converge to a single ancestor (i.e. they 'coalesce'). This fact can be used to derive the rate of gene fixation of a neutral allele (that is, one not under any form of selection) for a population of varying size (provided that it is finite and nonzero). Because the effect of natural selection is stipulated to be negligible, the probability at any given time that an allele will ultimately become fixed at its locus is simply its frequency $p$ in the population at that time. For example, if a population includes allele A with frequency equal to 20%, and allele a with frequency equal to 80%, there is an 80% chance that after an infinite number of generations a will be fixed at the locus (assuming genetic drift is the only operating evolutionary force).

For a diploid population of size N and neutral mutation rate $\mu$, the initial frequency of a novel mutation is simply 1/(2N), and the number of new mutations per generation is $2N\mu$. Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is $2N\mu \times \frac{1}{2N} = \mu$. Thus, the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations.[4][5]

Probability of fixation is also influenced by population size changes. For growing populations, selection coefficients are more effective. This means that beneficial alleles are more likely to become fixed, whereas deleterious alleles are more likely to be lost. In populations that are shrinking in size, selection coefficients are not as effective. Thus, there is a higher probability of beneficial alleles being lost and deleterious alleles being fixed. This is because if a beneficial mutation is rare, it can be lost purely due to chance of that individual not having offspring, no matter the selection coefficient. In growing populations, the average individual has a higher expected number of offspring, whereas in shrinking populations the average individual has a lower number of expected offspring. Thus, in growing populations it is more likely that the beneficial allele will be passed on to more individuals in the next generation. This continues until the allele flourishes in the population, and is eventually fixed. However, in a shrinking population it is more likely that the allele may not be passed on, simply because the parents produce no offspring. This would cause even a beneficial mutation to be lost.[6]

Time to fixation

Additionally, research has been done into the average time it takes for a neutrally selected gene to become fixed. Using mathematical treatments and computer based algorithms, Kimura and Ohta (1968) were able to discover that for a single mutated gene that is neutrally selected for, it takes an average of Ne generations until the mutation is fixed.[2] Here, Ne is the effective population size, or the number of individuals in an idealized population that would generate the same value of a population statistic that is observed for the real population. Usually the population statistic used to define effective population size is heterozygosity, but others can be used.[7]

Fixation rates can easily be modeled as well to see how long it takes for a gene to become fixed with varying population sizes and generations. For example, at The Biology Project Genetic Drift Simulation you can model genetic drift and see how quickly the gene for worm color goes to fixation in terms of generations for different population sizes.

References

1. ^ a b Arie Zackay (2007). Random Genetic Drift & Gene Fixation.
2. ^ a b Kimura, Motoo; Ohta, Tomoko (26 July 1968). "The average number of generations until fixation of a mutant gene in a finite population". Genetics 61: 763–771. Retrieved 12 September 2014.
3. ^ Kimura, Motoo (29 January 1962). "On the probability of fixation of mutant genes in a population". Genetics 47: 713–719. Retrieved 14 September 2014.
4. ^
5. ^
6. ^ Otto, Sarah; Whitlock, Michael (7 March 1997). "The probability of fixation in populations of changing size". Genetics 146: 723–733. Retrieved 14 September 2014.
7. ^ Caballero, Armando (9 March 1994). "Developments in the prediction of effective population size". Heredity 73: 657–679. Retrieved 12 September 2014.