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Figure 2.6 A De Finetti diagram for one locus with two alleles. The triangular coordinate system results from the requirement that the frequencies of all three genotypes must sum to one. Any point inside or on the edge of the triangle represents all three genotype frequencies of a population. The parabola describes Hardy-Weinberg expected genotype frequencies. The dashed lines represent the frequencies of each of the three genotypes between zero and one. Genotype frequencies at any point can be determined by the length of lines that are perpendicular to each of the sides of the triangle. A practical way to estimate genotype frequencies on the diagram is to hold a ruler parallel to one of the sides of the triangle and mark off the distance on one of the frequency axes. The point on the parabola is a population in Hardy-Weinberg equilibrium where the frequency of AA is 0.36, the frequency of aa is 0.16, and the frequency of Aa is 0.48. The perpendicular line to the base of the triangle also divides the bases into regions corresponding in length to the allele frequencies. Any population with genotype frequencies not on the parabola has an excess (above the parabola) or deficit (below the parabola) of heterozygotes compared to Hardy-Weinberg expected genotype frequencies.

a fair coin only a few times may not produce an equal number of heads and tails. Natural selection is a process that causes some genotypes in either the parental or progeny generations to be more frequent than others. So it is logical that Hardy-Weinberg expectations would not be met if natural selection were acting. In a sense, these assumptions define the biological processes that make up the field of population genetics. Each assumption represents one of the conceptual areas where population genetics can make testable predictions via expectation in order to distinguish the biological processes operating in populations. This is quite a set of accomplishments for an equation with just three terms!

Despite all of this praise, you might ask: what good is a model with so many restrictive assumptions? Are all these assumptions likely to be met in actual populations? The Hardy-Weinberg model is not necessarily meant to be an exact description of any actual population, although actual populations often exhibit genotype frequencies predicted by Hardy-Weinberg. Hardy-Weinberg provides a null model, a prediction based on a simplified or idealized situation where no biological processes are acting and genotype frequencies are the result of random combination. Actual populations can be compared with this null model to test hypotheses about the evolutionary forces acting on allele and genotype frequencies. The important point and the original motivation for Hardy and Weinberg was to show that the process of particulate inheritance itself does not cause any changes in allele frequencies across generations. Thus, changes in allele frequency or departures from Hardy-Weinberg expected genotype frequencies must be caused by processes that alter the outcome of basic inheritance.

These assumptions make intuitive sense when each is examined in detail (although this will probably be more apparent after more reading and simulation). As we will see later, Hardy-Weinberg holds for any number of alleles, although equation 2.1 is valid for only two alleles. Many of the assumptions can be thought of as assuring random mating and production of all possible progeny genotypes. Hardy-Weinberg genotype frequencies in progeny would not be realized if the two sexes have different allele frequencies even if matings take place between random pairs of parents. It is also possible that just by chance not all genotypes would be produced if only a small number of parents mated, just like flipping

Null model A testable model of no effect. A prediction or expectation based on the simplest assumptions to predict outcomes. Often, null models make predictions based on purely random processes, random samples, or variables having no effect on an outcome.

In the final part of this section we will explore genotype frequency expectations adjusted to account for ploidy (number of homologous chromosomes)

differences between males and females as seen in chromosomal sex determination and haplo-diploid organisms. In chromosomal sex determination as seen in mammals, birds, and Lepidoptera (butterflies and moths), one sex is determined by possession of two identical chromosomes (the homogametic sex) and the other sex determined by possession of two different chromosomes (the heterogametic sex). In mammals females are homogametic (XX) and males heterogametic (XY), whereas in birds and Lepidoptera the opposite is true, with heterogametic females (ZW) and homogametic males (ZZ). In haplo-diploid species such as bees and wasps (Hymenoptera), males are haploid (hemizygous) for all chromosomes whereas females are diploid for all chromosomes.

Predicting genotype frequencies at one locus in these cases under random mating and the other assumptions of Hardy-Weinberg requires keeping track of allele or genotype frequencies in both sexes and loci on specific chromosomes. An effective method is to draw a Punnett square that distinguishes the sex of an individual as well as the gamete types that can be generated at mating (Table 2.1). The Punnett square shows that genotype frequencies in the diploid sex are identical to Hardy-Weinberg expectations for autosomes, whereas genotype frequencies are equivalent to allele frequencies in the haploid sex. One consequence of different chromosome types between the sexes is that fully recessive phenotypes are more common in the heterogametic sex, where a single chromosome determines the phenotype and recessive phenotypes appear at the allele frequency. However, in the homogametic sex, fully recessive phenotypes appear at the frequency of the recessive genotype (e.g. q2) since they are masked in heterozygotes. Some types of color blindness in humans are examples of traits due to genes on the X chromosome (called "X-linked" traits) that are more common in men than in women due to haplo-diploid inheritance.

Later, in section 2.4, we will examine two categories of applications of Hardy-Weinberg expected genotype frequencies. The first set of applications arises when we assume (often with supporting evidence)

Table 2.1 Punnett square to predict genotype frequencies for loci on sex chromosomes and for all loci in males and females of haplo-diploid species. Notation in this table is based on birds where the sex chromosomes are Z and W (ZZ males and ZW females) with a diallelic locus on the Z chromosome possessing alleles A and a at frequencies p and q, respectively. In general, genotype frequencies in the homogametic or diploid sex are identical to Hardy-Weinberg expectations for autosomes, whereas genotype frequencies are equal to allele frequencies in the heterogametic or haploid sex.

Table 2.1 Punnett square to predict genotype frequencies for loci on sex chromosomes and for all loci in males and females of haplo-diploid species. Notation in this table is based on birds where the sex chromosomes are Z and W (ZZ males and ZW females) with a diallelic locus on the Z chromosome possessing alleles A and a at frequencies p and q, respectively. In general, genotype frequencies in the homogametic or diploid sex are identical to Hardy-Weinberg expectations for autosomes, whereas genotype frequencies are equal to allele frequencies in the heterogametic or haploid sex.

Hemizygous or haploid |
sex |

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