Fixation (population genetics)

Probability

Neutral alleles

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

Non-neutral alleles

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 non-overlapping generations n = 0, 1, 2, 3, ... , and with ${\displaystyle X_n}$ 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 ${\displaystyle X_0=1}$. The number of offspring of any one individual must follow a fixed distribution and is independently determined. In this framework the generating functions ${\displaystyle p_n(x)}$ for each ${\displaystyle X_n}$ satisfy the recursion relation ${\displaystyle p_n(x)=p_1(p_{n-1}(x))}$ and can be used to compute the probabilities ${\displaystyle {\pi }_n=P(X_n=0)}$ of no descendants at time n. It can be shown that ${\displaystyle {\pi }_n=p_1({\pi }_{n-1})}$, and furthermore, that the ${\displaystyle {\pi }_n}$ converge to a specific value ${\displaystyle \pi }$, which is the probability that the individual will have no descendants. The probability of fixation is then ${\displaystyle 1-\pi \approx 2s/{\sigma }^2}$ 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 (~${\displaystyle 1/N_e}$) and selection (~${\displaystyle s}$), where ${\displaystyle N_e}$ is the effective population size. The ratio ${\displaystyle N_{e}s}$ 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 ${\displaystyle N_e}$, a deleterious allele with selection coefficient ${\displaystyle -s}$ has a probability fixation equal to ${\displaystyle (1-e^{-2s})/(1-e^{-4N_{e}s})}$. This estimate can be obtained directly from Kimura's 1962 work. Deleterious alleles with selection coefficients ${\displaystyle -s}$ satisfying ${\displaystyle 2N_{e}s\ll 1}$ are effectively neutral, and consequently have a probability of fixation approximately equal to ${\displaystyle 1/2N_e}$.

Effect of growing/shrinking populations

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.

Time

Additionally, research has been done into the average time it takes for a neutral mutation to become fixed. Kimura and Ohta (1969) showed that a new mutation that eventually fixes will spend an average of 4N_e generations as a polymorphism in the population. Average time to fixation N_e is the effective population size, the number of individuals in an idealised population under genetic drift required to produce an equivalent amount of genetic diversity. Usually the population statistic used to define effective population size is heterozygosity, but others can be used.

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, The Biology Project Genetic Drift Simulation allows to model genetic drift and see how quickly the gene for worm color goes to fixation in terms of generations for different population sizes.

Additionally, fixation rates can be modeled using coalescent trees. A coalescent tree traces the descent of alleles of a gene in a population. It aims to trace back to a single ancestral copy called the most recent common ancestor.

Examples in research

In 1969, Schwartz at Indiana University was able to artificially induce gene fixation into maize, by subjecting samples to suboptimal conditions. Schwartz located a mutation in a gene called Adh1, which when homozygous causes maize to be unable to produce alcohol dehydrogenase. Schwartz then subjected seeds, with both normal alcohol dehydrogenase activity and no activity, to flooding conditions and observed whether the seeds were able to germinate or not. He found that when subjected to flooding, only seeds with alcohol dehydrogenase activity germinated. This ultimately caused gene fixation of the Adh1 wild type allele. The Adh1 mutation was lost in the experimented population.

In 2014, Lee, Langley, and Begun conducted another research study related to gene fixation. They focused on Drosophila melanogaster population data and the effects of genetic hitchhiking caused by selective sweeps. Genetic hitchhiking occurs when one allele is strongly selected for and driven to fixation. This causes the surrounding areas to also be driven to fixation, even though they are not being selected for. By looking at the Drosophila melanogaster population data, Lee et al. found a reduced amount of heterogeneity within 25 base pairs of focal substitutions. They accredit this to small-scale hitchhiking effects. They also found that neighboring fixations that changed amino acid polarities while maintaining the overall polarity of a protein were under stronger selection pressures. Additionally, they found that substitutions in slowly evolving genes were associated with stronger genetic hitchhiking effects.