**Hardy-Weinberg Equilibrium**

If we consider a pair of alleles *A* and *a*, we will find that the percentage of gametes in the gene pool bearing *A* or *a* will depend upon the genotypic frequencies of the parental generation whose gametes form the pool. For example, if most of the population have the recessive genotype *aa*, the frequency of the recessive allele in the gene pool would be relatively high, and the percentage of gametes bearing the dominant *A* allele would be correspondingly low.

When matings between members of a population are completely at random, i.e., when every male gamete in the gene pool has an equal opportunity of uniting with every female gamete, then the zygotic frequencies expected in the next generation may be predicted from a knowledge of the gene (allelic) frequencies in the gene pool of the parental population. That is, given the relative frequencies of *A* and *a* gametes in the gene pool, we can calculate (on the basis of the chance union of gametes) the expected frequencies of progeny genotypes and phenotypes. If *p* = percentage of *A* alleles in the gene pool and *q* = percentage of *a* alleles, then we can use the Punnett square to produce all the possible chance combinations of these gametes.

Note that *p* + *q* = 1; i.e., the percentage of *A* and *a* gametes must add to 100% in order to account for all of the gametes in the gene pool. The expected genotypic (zygotic) frequencies in the next generation are summarized as follows:

Thus, *p*^{2} is the fraction of the next generation expected to be homozygous dominant (*AA*), 2*pq* is the fraction expected to be heterozygous (*Aa*), and *q*^{2} is the fraction expected to be recessive (*aa*). These genotypic fractions must add to one (1.0) to account for all genotypes in the progeny population.

This formula, expressing the genotypic expectations of progeny in terms of the gametic (allelic) frequencies of the parental gene pool is called the **Hardy-Weinberg** rule, named after G. H. Hardy and W. Weinberg who independently formulated it in 1908. If a population conforms to the conditions on which this formula is based, there should be no change in the allele frequencies in the population from generation to generation. Should a population initially be in disequilibrium, one generation of random mating is sufficient to bring it into genetic equilibrium, and thereafter the population will remain in equilibrium (unchanging in allelic frequencies) as long as the Hardy-Weinberg conditions persist.

**Assumptions and Restrictions**

Several assumptions underlie the attainment of genetic equilibrium as expressed in the Hardy-Weinberg equation. When these conditions are met, the predictions from the rule are valid.

- The population is infinitely large and mates at random (
**panmictic**). - No selection is operative, i.e., each genotype under consideration can survive just as well as any other (no differential mortality), and each genotype is equally efficient in the production of progeny (no differential reproduction).
- The population is closed, i.e., no immigration of individuals from another population into nor emigration from the population under consideration is allowed.
- There is no mutation from one allelic state to another. Mutation may be allowed if the forward and back mutation rates are equivalent, i.e.,
*A*mutates to*a*with the same frequency that*a*mutates to*A*. - Meiosis is normal so that chance is the only factor operative in gameto-genesis.

If we define **evolution** as any change in the allele frequencies of a population, then a violation of one or more of the Hardy-Weinberg restrictions could cause the population to move away from the equilibrium frequencies and thus evolve. Changes in gene frequencies can be produced by sampling errors most evident in a very small populations (**genetic drift**), by selection, migration, or mutation pressures, or by nonrandom assortment of chromosomes (**meiotic drive**). No population is infinitely large, spontaneous mutations cannot be prevented, selection and migration pressures usually exist in most natural populations, etc., so it may be surprising to learn that despite these violations of Hardy-Weinberg restrictions many populations do conform, within statistically acceptable limits, to equilibrium conditions between two successive generations. Changes too small to be statistically significant deviations from equilibrium expectations between any two generations can nonetheless accumulate over many generations to produce considerable alterations in the genetic structure of a population.

A **race** is a phenotypically, genetically, and usually geographically isolated interbreeding population of a species. Races of a given species can freely interbreed with one another. Members of different **species**, however, are generally reproductively isolated to a recognizable degree. **Subspecies** are races that have been given distinctive taxonomic names. Varieties, breeds, strains, etc., of cultivated plants or domesticated animals may also be equated with the racial concept. Geographic isolation is usually required for populations of a species to become distinctive races. Race formation is generally a prerequisite to the splitting of one species into two or more species (**speciation**). Differentiation at many loci over many generations is generally required to reproductively isolate these groups by time of breeding, behavioral differences, ecological requirements, hybrid inviability, hybrid sterility, and other such mechanisms. Yet, clearly, single loci that determine traits significant for mate choice or recognition could yield a new species very rapidly.

Although alleles at a single autosomal locus reach equilibrium following one generation of random mating, gametic equilibrium involving two independently assorting genes is approached rapidly over a number of generations. At equilibrium, the product of coupling gametes equals the product of repulsion gametes.

EXAMPLE 9.1Consider one locus with allelesAandaat frequencies represented bypandq, respectively. A second locus has allelesBandbat frequenciesrands, respectively. The expected frequencies of coupling gametesABandabareprandqs, respectively. The expected frequencies of repulsion gametesAbandaBarepsandqr, respectively. At equilibrium, (pr) (qs) = (ps) (qr). Also at quilibrium, the disequilibrium coefficient (d) isd= (pr) (qs) – (ps) (qr) = 0.

For independently assorting loci under random mating, the disequilibrium value of *d* is halved in each generation during the approach to equilibrium because unlinked genes experience 50% recombination. The approach to equilibrium by linked genes, however, is slowed by comparison because they recombine less frequently than unlinked genes (i.e., less than 50% recombination). The closer the linkage, the longer it takes to reach equilibrium. The disequilibrium (*d*_{t}) that exists at any generation (*t*) is expressed as

d_{t}= (1 –r)d_{t–1}

where *r* frequency of recombination and *d*_{t–1} = disequilibrium in the previous generation.

EXAMPLE 9.2Ifd= 0.25 initially and the two loci experience 20% recombination (i.e., the loci are 20 map units apart), the disequilibrium that would be expected after one generation of random mating isdt= (1 – 0.2)(0.25)= 0.2. This represents 0.20/0.25 = 0.8 or 80% of the maximum disequilibrium that could exist for a pair of linked loci.

Practice problems for these concepts can be found at:

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