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. That is, the allele becomes fixed. ^[1] In the absence of mutation or heterozygote advantage, any allele must eventually either be lost completely from the population, or fixed, i.e. permanently established at 100% frequency in the population.^[2] Whether a gene will ultimately be lost or 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 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.

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Voiceover: What I want to do with this video is explore the idea of allele frequency. Allele frequency. Just as a reminder, an allele is a variant of a gene. You get a variant of a gene from your mother, and you get another variant of the gene from the father. So, when we're talking about the allele, we're talking about that specific variant that you got from your mother or your father. We've seen this before, but now let's dig a little bit deeper. To help us get our heads around this, we'll start with a fairly common model for this. We're going to think about eye color. Obviously, this is a very large simplification, but let's just assume that we have a population where there's only two variants of an eye color gene. Let's first assume there is an eye color gene. Let's assume there's two variants. One variant, one allele for eye color, we'll use the shorthand, capital B. Let's say that's the allele for brown. Brown eye color. We're going to assume that this one is dominant. It's dominant over the other allele. Now the other allele, we're going to assume is for blue eye color, and we'll represent that with a lower case B. So that is blue eye color, and we're going to assume that this is recessive. Once again, this is review. Someone who has one of the big B alleles, the brown alleles, it doesn't matter what their other allele is going to be, because it's either going to be another brown or it's going to be a blue, they're going to show brown eyes. This is going to be brown eyes, and this is going to be brown eyes, because the capital B is dominant. The only way to get blue eyes is to be a homozygote for the recessive allele. All of that, of course, is review. We've seen that before. Now let's think about allele frequency. To think about that, I'll set up a very artificially small population. Let's say our population has exactly two people in it. Population has exactly two people in it, Person 1 and Person 2, and let's say we're able to look into their DNA and figure out their genotypes. Person 1, say, has a capital B allele, has a brown allele and a blue allele, while Person 2 has two blue, two blue alleles. Given that we know the genotypes in this artificially small population, we could start thinking about the allele frequencies. Or the frequencies of the different alleles. What do you think is going to be the frequency the frequency of the brown allele in this population? I encourage you to pause this video and think about this on your own. I'm assuming you've had a go at it, so you might be tempted to say, "Looks like one out of two people "have it, maybe it's 50%." But that wouldn't be the right way to think about allele frequencies. In allele frequencies, you want to dig a little bit deeper and look at the individual alleles. When you look at that, you say, "Okay, there's "four individual alleles in this population, or there's "four variants, or there's literally four chromosomes "that are carrying that gene in this population." Out of them, one of them carry, one of them is the capital B allele, so we could say that that is going to be zero point two five, or 25%. Once again, 25% of the genes for eye color have the capital B allele, have the brown allele. Now we can do the same, ask ourselves the same question for the lower case B allele. What fraction of the genes in this population are code for or represent the lower case B, the blue allele? Once again, I encourage you to pause the video and think about it. Well, very similar idea. There's four genes in the population that are coding for eye color. Of them, one, two, three code for or are the lower case blue allele. So that's zero point seven five or 75%. 75% of the genes code for the lower case blue allele, while 25 are the brown allele. I really want to hit this point home, how this is different than, say, the phenotype frequency. If I asked you, in the population, if I asked you the percent of brown-eyed people, so now I'm talking about phenotype, what would that be? Well, there's two people in the population. One of them is exhibiting brown eyes, so that's going to be one-half. Similarly, if I were to ask you what is the percentage of people who are blue-eyed that, too, would be one-half. This person is one of the two people, they're exhibiting blue eyes. But allele frequency, we're digging deeper, we're looking at the genotypes. We're saying out of the four genes here, one of them is the big B allele, so that's 25% of the gene population codes for the brown allele and 75% is the blue allele. This is really important to internalize. Because once we internalize this, then as we'll see, that the ideas in the Hardy-Weinberg principle start to make a lot of sense. I'll do a little bit of foreshadowing. We can denote this, this is just a convention that's often used, by the lower case letter P, and we can use lower case Q to denote the frequency. So lower case P is the frequency of the dominant allele, lower case Q the frequency of the recessive allele. What's true here? What's true of P, what's going to be true of P plus Q, what's P plus Q going to be equal to? I encourage you to pause the video again and think about that. What is this going to be equal to? Well, when we started off, we said that there's only two potential, that's one of the assumptions we assumed, we assumed there's only two alleles in this population, in kind of the allele population for this gene population for this trait. The frequency of the dominant ones plus the frequency of recessive ones, well everyone's going to have one of those two, so if you add those two frequencies, it's going to have to add to 100%. We see that there. One-fourth plus three-fourths is one, or 100%. And 25% plus 75% is also 100%. So we could say P plus Q is equal to 100%, or we could say that P plus Q is equal to one. Is equal to one. So, in the next video, we're going to start from the seemingly fairly simple idea, to get to a more richer and fairly neat idea that's expressed in the Hardy-Weinberg equation.

History

The earliest mention of gene fixation in published works was found in Motoo 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.^[4]

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 $p$ 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.^[5]

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

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.^[6]

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.^[2] 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.^[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.

Additionally, fixation rates can be modeled using coalescent trees. A coalescent tree traces the descent of alleles of a gene in a population.^[8] It aims to trace back to a single ancestral copy called the most recent common ancestor.^[9]

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.^[10]

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.^[11] 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.^[12]

References

^ ^a ^b Arie Zackay (2007). Random Genetic Drift & Gene Fixation (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2013-08-29.
^ ^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 (3): 763–771. doi:10.1093/genetics/61.3.763. PMC 1212239. PMID 17248440.
^ Kimura, Motoo (1983). The Neutral Theory of Molecular Evolution. The Edinburgh Building, Cambridge: Cambridge University Press. ISBN 978-0-521-23109-1. Retrieved 16 November 2014.
^ ^a ^b Kimura, Motoo (29 January 1962). "On the probability of fixation of mutant genes in a population". Genetics. 47 (6): 713–719. doi:10.1093/genetics/47.6.713. PMC 1210364. PMID 14456043.
^ David H.A. Fitch (1997). Deviations from the null hypotheses: Finite populations sizes and genetic drift, mutation and gene flow.
^ Otto, Sarah; Whitlock, Michael (7 March 1997). "The probability of fixation in populations of changing size" (PDF). Genetics. 146 (2): 723–733. doi:10.1093/genetics/146.2.723. PMC 1208011. PMID 9178020. Retrieved 14 September 2014.
^ Caballero, Armando (9 March 1994). "Developments in the prediction of effective population size". Heredity. 73 (6): 657–679. doi:10.1038/hdy.1994.174. PMID 7814264.
^ Griffiths, RC; Tavare, Simon (1998). "The Age of a Mutation in a General Coalescent Tree". Communications in Statistics. Stochastic Models. 14 (1&2): 273–295. doi:10.1080/15326349808807471.
^ Walsh, Bruce (22 March 2001). "Estimating the Time to the Most Recent Common Ancestor for the Y chromosome or Mitochondrial DNA for a Pair of Individuals". Genetics. 158 (2): 897–912. doi:10.1093/genetics/158.2.897. PMC 1461668. PMID 11404350.
^ Schwartz, Drew (1969). "An Example of Gene Fixation Resulting from Selective Advantage in Suboptimal Conditions". The American Naturalist. 103 (933): 479–481. doi:10.1086/282615. JSTOR 2459409. S2CID 85366302.
^ Rice, William (12 February 1987). "Genetic Hitchhiking and the Evolution of Reduced Genetic Activity of the Y Sex Chromosome". Genetics. 116 (1): 161–167. doi:10.1093/genetics/116.1.161. PMC 1203114. PMID 3596229.
^ Lee, Yuh; Langley, Charles; Begun, David (2014). "Differential Strengths of Positive Selection Revealed by Hitchhiking Effects at Small Physical Scales in Drosophila melanogaster". Molecular Biology and Evolution. 31 (4): 804–816. doi:10.1093/molbev/mst270. PMC 4043186. PMID 24361994. Retrieved 16 November 2014.

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