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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.

      Null model: A testable model of no effect or a background effect. A prediction or expectation based on the simplest assumptions to predict outcomes. Often, population genetic null models make predictions based on purely random processes such as random mating or genetic drift, random samples or combinations, or variables having background effects on allele or genotype frequencies.

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

Table 2.1 Punnett square to predict genotype frequencies for loci on sex chromosomes and for all loci in males and females of haploid‐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, while genotype frequencies are equal to allele frequencies in the homogametic or haploid sex.

Homozygotic or diploid sex
			Genotype	ZZ
			Gamete	Z‐A Z‐a
Heterozygotic or haploid sex			Frequency	p q
Genotype	Gamete	Frequency
	Z‐A	p		Z‐A Z‐A	Z‐A Z‐a
				p 2	pq
ZW	Z‐a	q		Z‐A Z‐a	Z‐a Z‐a
				Pq	q 2
	W			Z‐A W	Z‐a W
				p	q
Expected genotype frequencies under random mating
Homogametic sex			Homogametic sex
Z‐A Z‐A	p 2		Z‐A W		p
Z‐A Z‐a
Z‐a Z‐a	q 2		Z‐a W		q

      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) that the assumptions of Hardy–Weinberg are true. We can then compare several expectations for genotype frequencies with actual genotype frequencies to distinguish between several alternative hypotheses. The second type of application is where we examine what results when assumptions of Hardy–Weinberg are not met. There are many cases where population genotype frequencies can be used to reveal the action of various population genetic processes. Before that, the next section builds a proof of the Hardy–Weinberg prediction that inheritance per se will not alter allele frequencies.

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