# Hardy-Weinberg law

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An understanding of evolution depends upon knowledge of population genetics. A population is a group of individuals of the same species in a given area whose members can interbreed. Because the individuals of a population can interbreed, they share a common group of genes known as the gene pool. The biological sciences now generally define evolution as being "the sum total of the genetically inherited changes in the individuals who are the members of a population's gene pool" . It is clear that the effects of evolution are felt by individuals, but it is the population as a whole that actually evolves. The gene frequency of an allele is the number of times an allele for a particular trait occurs compared to the total number of alleles for that trait. For evolution to occur in real populations, some of the gene frequencies must change with time. The Hardy–Weinberg law provides a baseline to determine whether or not gene frequencies have changed in a population, and thus whether evolution has occurred.

As a result of independent work in the early 20th century, G.H. Hardy, an English mathematician [a1], and W. Weinberg, a German physician [a3], concluded that gene pool frequencies are inherently stable but that evolution should be expected in all populations virtually all of the time. They resolved this apparent paradox by analyzing the probable net effects of evolutionary mechanisms using mathematical models.

An important way of discovering why real populations change with time is to construct a model of a population that does not change. This is just what Hardy and Weinberg did. Their principle describes a hypothetical situation in which there is no change in the gene pool (frequencies of alleles), hence no evolution.

Consider a population whose gene pool contains the alleles $A$ and $a$. Hardy and Weinberg assigned the letter $p$ to the frequency of the dominant allele $A$ and the letter $q$ to the frequency of the recessive allele $a$. In other words, $p$ equals all of the alleles in individuals who are homozygous dominant ($AA$) and half of the alleles in people who are heterozygous ($Aa$) for this trait. Likewise, $q$ equals all of the alleles in individuals who are homozygous recessive ($aa$) and the other half of the alleles in people who are heterozygous ($Aa$). In mathematical terms, these are

$$p=AA+\frac12Aa,q=aa+\frac12Aa.$$

Since the sum of all the alleles must equal $100$%, then $p+q=1$. They then reasoned that all the random possible combinations of the members of a population would equal $(p+q)^2=p^2+2pq+q^2=1$. This has become known as the Hardy–Weinberg equilibrium equation. In this equation, $p^2$ is the frequency of homozygous dominant ($AA$) people in a population, $2pq$ is the frequency of heterozygous ($Aa$) people, and $q^2$ is the frequency of homozygous recessive ($aa$) ones.

The Hardy–Weinberg equation can be used to discover the genotype frequencies in a population and to track their changes from one generation to another. From observations of phenotypes, it is usually only possible to know the frequency of homozygous recessive people, or $q^2$ in the equation, since they will not have the trait. Those who express the trait in their phenotype could be either homozygous dominant $p^2$ or heterozygous $2pq$. The Hardy–Weinberg equation allows one to determine which ones they are. Since $p=1-q$ and $q$ is known, it is possible to calculate $p$ as well. Knowing $p$ and $q$, it is a simple matter to plug these values into the Hardy–Weinberg equilibrium equation. This then provides the frequencies of all three genotypes for the selected trait within the population. Using phenotype frequencies from the next generation in a population, one can also learn whether or not evolution has occurred and in what direction and rate for the selected trait. However, the Hardy–Weinberg equation cannot determine which of the various possible causes of evolution were responsible for the changes in gene pool frequencies.

Hardy, Weinberg, and the population geneticists who followed them came to understand that evolution will not occur in a population if seven conditions are met:

1) the population is infinitely large;

2) all mating is totally random;

3) mutation is not occurring;

4) natural selection is not occurring;

5) there is no migration in or out of the population;

6) all members of the population breed; and

7) everyone produces the same number of offspring. So long as the above conditions are met, gene frequencies and genotype ratios in a randomly-breeding population remain constant from generation to generation. In other words, if no mechanisms that can cause evolution to occur are acting on a population, evolution will not occur — the gene pool frequencies will remain unchanged. However, since it is highly unlikely that any of these seven conditions, let alone all of them, will occur in the real world, evolution is the inevitable result. What the law expresses is that populations are able to maintain a reservoir of variability so that if future conditions require it, the gene pool can change. If recessive alleles were continually tending to disappear, the population would soon become homozygous. Under Hardy–Weinberg conditions, genes that have no present selective value will nonetheless be retained.

#### References

 [a1] G.H. Hardy, "Mendelian proportions in a mixed population" Sci. , 28 (1908) pp. 49–50 [a2] C. Stren, "The Hardy–Weinberg law" Sci. , 97 (1943) pp. 137–138 [a3] W. Weinberg, "On the demonstration of heredity in man" , Papers on Human Genetics , Prentice-Hall (1963) (Original: 1980; Translation by S.H. Boyer)
How to Cite This Entry:
Hardy–Weinberg law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy%E2%80%93Weinberg_law&oldid=22556