Fixation (population genetics)

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) in a given population to a situation where only one of the alleles remains. In the absence of mutation or heterozygote advantage, any allele must eventually be lost completely from the population or fixed (permanently established at 100% frequency in the population). Whether a gene will ultimately be lost or fixed is dependent on selection coefficients and chance fluctuations in allelic proportions. Fixation can refer to a gene in general or particular nucleotide position in the DNA chain (locus). 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) in a given population to a situation where only one of the alleles remains. In the absence of mutation or heterozygote advantage, any allele must eventually be lost completely from the population or fixed (permanently established at 100% frequency in the population). Whether a gene will ultimately be lost or fixed is dependent on selection coefficients and chance fluctuations in allelic proportions. 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 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. Similarly, genetic differences between taxa are said to have been fixed in each species. The earliest mention of gene fixation in published works was found in Kimura's 1962 paper 'On Probability of Fixation of Mutant Genes in a Population'. In the paper, Kimura uses mathematical techniques to determine the probability of fixation of mutant genes in a population. He showed that the probability of fixation depends on the initial frequency of the allele and the mean and variance of the gene frequency change per generation. 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 {displaystyle 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 μ {displaystyle mu } , the initial frequency of a novel mutation is simply 1/(2N), and the number of new mutations per generation is 2 N μ {displaystyle 2Nmu } . Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is 2 N μ × 1 2 N = μ {displaystyle 2Nmu imes {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. For fixed population sizes, the probability of fixation for a new allele with selective advantage s can be approximated using the theory of branching processes. A population with nonoverlapping generations n = 0, 1, 2, 3, ... , and with X n {displaystyle X_{n}} genes (or 'individuals') at time n forms a Markov chain under the following assumptions. The introduction of an individual possessing an allele with a selective advantage corresponds to X 0 = 1 {displaystyle X_{0}=1} . The number of offspring of any one individual must follow a fixed distribution and is independently determined. In this framework the generating functions p n ( x ) {displaystyle p_{n}(x)} for each X n {displaystyle X_{n}} satisfy the recursion relation p n ( x ) = p 1 ( p n − 1 ( x ) ) {displaystyle p_{n}(x)=p_{1}(p_{n-1}(x))} and can be used to compute the probabilities π n = P ( X n = 0 ) {displaystyle pi _{n}=P(X_{n}=0)} of no descendants at time n. It can be shown that π n = p 1 ( π n − 1 ) {displaystyle pi _{n}=p_{1}(pi _{n-1})} , and furthermore, that the π n {displaystyle pi _{n}} converge to a specific value π {displaystyle pi } , which is the probability that the individual will have no descendants. The probability of fixation is then 1 − π ≈ 2 s / σ 2 {displaystyle 1-pi approx 2s/sigma ^{2}} since the indefinite survival of the beneficial allele will permit its increase in frequency to a point where selective forces will ensure fixation. Weakly deleterious mutations can fix in smaller populations through chance, and the probability of fixation will depend on rates of drift (~ 1 / N e {displaystyle 1/N_{e}} ) and selection (~ s {displaystyle s} ), where N e {displaystyle N_{e}} is the effective population size. The ratio N e s {displaystyle N_{e}s} determines whether selection or drift dominates, and as long as this ratio is not too negative, there will be an appreciable chance that a mildly deleterious allele will fix. For example, in a diploid population of size N e {displaystyle N_{e}} , a deleterious allele with selection coefficient − s {displaystyle -s} has a probability fixation equal to ( 1 − e − 2 s ) / ( 1 − e − 4 N e s ) {displaystyle (1-e^{-2s})/(1-e^{-4N_{e}s})} . This estimate can be obtained directly from Kimura's 1962 work. Deleterious alleles with selection coefficients − s {displaystyle -s} satisfying 2 N e s ≪ 1 {displaystyle 2N_{e}sll 1} are effectively neutral, and consequently have a probability of fixation approximately equal to 1 / 2 N e {displaystyle 1/2N_{e}} . 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.