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The Hardy-Weinberg principle and its applications in modern population genetics

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The discovery of the Hardy-Weinberg principle marked the beginning of the field of population genetics.

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Viability selection influences the genotypic contexts of alleles and leads to quantifiable departures from Hardy-Weinberg proportions. One measure of these departures is Wright's inbreeding coefficient F , where observed heterozygosity is compared with expected heterozygosity. Here, I extend population genetics theory to describe post-selection genotype frequencies in terms of post-selection allele frequencies and fitness dominance. The resulting equations correspond to non-equilibrium populations, allowing the following questions to be addressed: When selection is present, how large a sample size is needed to detect significant departures from Hardy-Weinberg?

How do selection-induced departures from Hardy-Weinberg vary with allele frequencies and levels of fitness dominance? For realistic selection coefficients, large sample sizes are required and departures from Hardy-Weinberg proportions are small.

Natural selection modifies the probabilities that alleles are found in either homozygous or heterozygous form. Given that one allele is A , what is the probability that the homologous copy of this gene is also A? In Hardy-Weinberg populations this is simply equal to p , the allele frequency of the A allele.

When the assumptions of the Hardy-Weinberg principle are violated, such as when viability selection is present, this result cannot be expected to hold. While this has been known for decades, many current studies assume Hardy-Weinberg proportions p 2 : 2 pq : q 2 without explicitly considering the impact of selection. When viability selection results in significant departures from Hardy Weinberg DHW , the genetic footprint of natural selection can be observed in sequence data [ 1 - 3 ].

Tests of Hardy-Weinberg proportions have been used to detect genotyping errors [ 4 - 6 ]. However, it is an open question whether natural selection confounds such tests. Consequently, one can ask: When does natural selection result in significant departures from Hardy-Weinberg proportions? For example, one expects to only find post-selection copies of a recessive lethal in heterozygotes. While equations describing genotypic frequencies in terms of allele frequencies are deducible for overdominance, mutation-selection balance, and other equilibria, existing theory is lacking when it comes to non-equilibrium populations [ 8 ].

There is a need to determine when viability selection leads to significant departures from Hardy-Weinberg proportions [ 9 ]. Classical population genetics contains recursion equations that describe post-selection genotype frequencies in terms of pre-selection allele frequencies.

However, DHW calculations require allele and genotype frequencies to be from the same time point i. In this paper population genetics theory is extended, and novel equations are derived for non-equilibrium populations at a single time point. These equations allow the magnitude of viability selection-induced DHW to be quantified and statistical significance to be assessed.

A number of statistical tests of Hardy-Weinberg proportions exist [ 10 - 13 ]. However, these tests do not distinguish between different causes of DHW such as genetic drift, population subdivision, genotyping error, and natural selection. By coupling population genetics theory to tests from statistical genetics one can determine whether observed departures from Hardy-Weinberg are due to selection.

Sample sizes needed to detect selection are found, and they are substantial. A classical population genetics model is used: Hardy-Weinberg plus selection.

Consider a single locus with two segregating alleles. Assume that mutation rates are negligible, and generations are discrete and non-overlapping. The population is assumed to be panmictic and large, yielding a deterministic model. Viability selection acts upon zygotes prior to adulthood, with constant genotypic fitnesses denoted by w AA , w AB , and w BB.

Allele frequencies are represented by lower case letters, with pre-selection allele frequencies in boldface p and q and post-selection allele frequencies in normal typeface p and q. After random mating, genotype frequencies are found in Hardy-Weinberg proportions.

Genotype frequencies are subsequently weighted by fitness, resulting in the following classic equations from population genetics:. The above equations can be algebraically manipulated, yielding an equality that contains only post-selection genotype frequencies [ 14 ]. Post-selection genotype frequencies are mathematically related to genotype fitnesses [ 15 ], and the ratio of genotypic fitnesses in the right hand side of equation 2 can be replaced by a single parameter that represents the extent of fitness dominance k.

Note that k is always positive. Post-selection genotype frequencies differ from Hardy-Weinberg expectations. As per classical population genetics: genotype frequencies sum to one, and allele frequencies are simply weighted genotypic frequencies.

These properties, in addition to equation 2 , can be combined to obtain post-selection genotype frequencies as a function of post-selection allele frequencies p and the ratio of genotypic fitnesses k. Factoring with respect to P AB produces a second order polynomial equation:.

For all possible values of p and k , the discriminant is positive i. However, only one root of the quadratic equation produces valid genotype frequencies.

The positive root of the quadratic equation results in heterozygote frequencies between zero and one see equation 6 below. The equations below reduce the description of a post-selection population genetic state to a single allele frequency rather than a collection of genotype frequencies. Using the above equations, the magnitude of viability selection-induced DHW can be quantified.

Multiple measures of DHW exist, with one common measure being Wright's inbreeding coefficient [ 3 , 16 ]. This is equal to one minus the observed heterozygosity over expected heterozygosity. Note that genotype and allele frequencies in equation 8 are all post-selection. When F is negative there is an excess of heterozygotes, and when F is positive there is a deficit of heterozygotes relative to Hardy-Weinberg expectations.

Just as inbreeding can lead to DHW, so too can natural selection. Let F sel be a measure of selection-induced DHW. F sel is derived from equations 6 and 8 :. Genotype frequencies in a sample of size n need not equal the true genotype frequencies of a population. Given a sample of size n , the test statistic X 2 can be calculated.

If sample size is large, X 2 is conveniently related to F [ 17 ]. When a null hypothesis of Hardy-Weinberg proportions is true, X 2 is approximately distributed as a chi-square with one degree of freedom. When a null hypothesis of Hardy-Weinberg proportions is false, X 2 is approximately distributed as a non-central chi-square [ 17 ]. Consequently, equation 11 can be rearranged to yield the sample size required to detect selection at a significance level of 0.

The sign and magnitude of selection-induced departures from Hardy-Weinberg are determined by allele frequencies and fitness dominance. Departures from Hardy-Weinberg can be measured by an inbreeding coefficient F sel. Note that while F-statistics are used, this does not imply that any actual inbreeding is present.

DHW due to viability selection is maximized at intermediate allele frequencies, and minimized when one allele is rare. This is because inbreeding coefficients are relatively insensitive to DHW when minor allele frequencies are close to zero.

When k takes on intermediate values i. The magnitude of selection-induced departures from Hardy-Weinberg proportions. F sel is a function of allele frequency p and fitness dominance k ; negative values of F sel indicate an excess of heterozygotes, while positive values of F sel indicate a deficit of heterozygotes, the dashed line corresponds to Hardy-Weinberg proportions. To detect selection, sample sizes ranging from thousands to millions are required.

Statistical significance is set at 0. Note that small sample sizes are more likely to result in observed allele frequencies that differ from the true allele frequencies of a population. When selection coefficients are small, even larger sample sizes are needed. Weak selection and unequal allele frequencies require larger sample sizes, while strong selection and equal allele frequencies require smaller sample sizes. When alleles are found at intermediate frequencies, required sample sizes are largely independent of p.

Here, sample genotype frequencies were drawn via multinomial sampling and tested for significant DHW. This was done times for each set of parameters, and observed power closely matched expected power. Sample size as a function of allele frequency and fitness dominance. Sample sizes n required to detect selection at a significance level of 0.

Sample sizes were obtained from equations 11 and 12 ; for each parameter set, true post-selection genotype frequencies were obtained from equations 5 , 6 , and 7 ; sample genotype counts were then generated via multinomial sampling, and chi-square tests were performed; MATLAB simulations were run times for each parameter set, and the proportion of tests that resulted in detectable DHW were recorded.

For moderate levels of fitness dominance i. Consequently, Hardy-Weinberg proportions reasonably approximate post-selection genotype frequencies. An interesting property of Hardy-Weinberg Equilibrium is that one can infer complete single-locus genotypic states from partial data i. This also holds for post-selection frequencies in a one-locus, two-allele system.

An exception involves heterozygote frequency data which maps to a pair of possible allele frequencies. Given genotypic fitnesses and single genotype frequency, p can be found via equation 5 , 6 , or 7.

Subsequently, p and k can be used to obtain the post-selection frequencies of other genotypes. In practice, however, one is much more likely to have complete genotype frequency data than complete knowledge of genotypic fitnesses. Statistically significant DHW requires large departures from neutrality and is maximized at intermediate allele frequencies. It is known that non-central chi-square tests can over-estimate statistical power when alternative hypotheses differ greatly in their expectations [ 20 ].

However, selection-induced departures from Hardy-Weinberg proportions are of small magnitude. If only two alleles are segregating, heterozygosity tests of neutrality require large sample sizes [ 21 , 22 ]. Many alleles are nearly neutral [ 23 ], with values of k close to one. However, the scope of undetectable selection extends over a much wider range of parameter space than the range of nearly neutral genes.

DHW is a poor indicator of natural selection in the wild. This qualitative conclusion is unlikely to be changed when the assumptions of this paper's model are relaxed. Mutation, assortative mating, and finite population size are all likely to further obscure the signature of selection on genotype frequencies.

Also note that genes under directional selection are less likely to be observed at intermediate allele frequencies i. A lack of significant DHW does not imply neutrality. There are large regions of parameter space where viability selection can lead notable changes in allele frequencies over time without producing significant DHW in any single generation. Multiple mechanisms can result in a failure to detect selection even when it is present i.

The Hardy-Weinberg principle and its applications in modern population genetics

The frequencies of genotypes and of alleles have a relationship. When the allele frequencies are known, the genotype frequencies can be calculated. This relationship is known in population genetics as the law of Hardy and Weinberg. This law is valid in a population, stable e. In such a stable population the Hardy and Weinberg equilibrium is at stake. Hardy and Weinberg equilibrium implies that in large populations with random mating among parents, and in the absence of selection, migration, mutation and random drift, the genotype and allele frequencies are constant do not change from generation to generation and the genotype frequencies can be calculated from the allele frequencies. The Hardy and Weinberg equilibrium indicates the stability of a population over generations.

This page has been archived and is no longer updated. Most populations have some degree of variation in their gene pools. By measuring the amount of genetic variation in a population, scientists can begin to make predictions about how genetic variation changes over time. These predictions can then help them gain important insights into the processes that allow organisms to adapt to their environment or to develop into new species over generations, also known as the process of evolution. Genetic variation is usually expressed as a relative frequency, which means a proportion of the total population under study. In other words, a relative frequency value represents the percentage of a given phenotype, genotype, or allele within a population.

Viability selection influences the genotypic contexts of alleles and leads to quantifiable departures from Hardy-Weinberg proportions. One measure of these departures is Wright's inbreeding coefficient F , where observed heterozygosity is compared with expected heterozygosity. Here, I extend population genetics theory to describe post-selection genotype frequencies in terms of post-selection allele frequencies and fitness dominance. The resulting equations correspond to non-equilibrium populations, allowing the following questions to be addressed: When selection is present, how large a sample size is needed to detect significant departures from Hardy-Weinberg? How do selection-induced departures from Hardy-Weinberg vary with allele frequencies and levels of fitness dominance?


The Hardy-Weinberg principle applies to individual genes with two alleles, a dominant allele and a recessive allele. A population with such a gene can be.


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In population genetics , the Hardy—Weinberg principle , also known as the Hardy—Weinberg equilibrium, model, theorem , or law , states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include genetic drift , mate choice , assortative mating , natural selection , sexual selection , mutation , gene flow , meiotic drive , genetic hitchhiking , population bottleneck , founder effect and inbreeding. In the absence of selection, mutation, genetic drift, or other forces, allele frequencies p and q are constant between generations, so equilibrium is reached.

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The Hardy-Weinberg principle and its applications in modern population genetics

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