Hardy-Weinberg Law Of Genetic Equilibrium

Genetics began with the study of inheritance in families: Gregor Mendel’s laws describe how the alleles of a pair of individuals are distributed among their offspring. Population genetics is the branch of genetics that studies the behavior of genes in populations. The population is the only biological unit that can persist for a span of time greater than the life of an individual, and the population is the only biological unit that can evolve. The two main subfields of population genetics are theoretical (or mathematical) population genetics, which uses formal analysis of the properties of ideal populations, and experimental population genetics, which examines the behavior of real genes in natural or laboratory populations.

Population genetics began as an attempt to extend Mendel’s laws of inheritance to populations. In 1908, Godfrey H. Hardy, an English mathematician, and Wilhelm Weinberg, a German physician, each independently derived a description of the behavior of allele and genotype frequencies in an ideal population of sexually reproducing diploid organisms. Their results, now termed the Hardy-Weinberg principle, or Hardy-Weinberg equilibrium, showed that the pattern of allele and genotype frequencies in such a population followed simple rules. They also showed that, in the absence of external pressures for change, the genetic makeup of a population will remain the same, at an equilibrium. Since evolution is the change in a population over time, such a population is not evolving. Modern evolutionary theory is an outgrowth of the “New Synthesis” of R. A. Fisher, J. B. S. Haldane, and Sewall Wright, which was done in the 1930s. They examined the significance of various factors that cause evolution by examining the degree to which they cause deviations from the predictions of the Hardy-Weinberg equilibrium.

Assumptions and Predictions

The predictions of the Hardy-Weinberg equilibrium hold if the following assumptions are true: The population is infinitely large; there is no differential movement of alleles or genotypes into or out of the population; there is no mutation (no new alleles are added to the population); there is random mating (all genotypes have an equal chance of mating with all other genotypes); and all genotypes are equally fit (have an equal chance of surviving to reproduce). Under this very restricted set of assumptions, the following two predictions are true—Allele frequencies will not change from one generation to the next, and genotype frequencies can be determined by a simple equation and will not change from one generation to the next.

The predictions of the Hardy-Weinberg equilibrium represent the working through of a simple set of algebraic equations and can be easily extended to more than two alleles of a gene. In fact, the results were so self-evident to the mathematician Hardy that he at first did not think the work was worth publishing.

If there are two alleles (A, a) for a gene present in the gene pool, let p = the frequency of the A allele and q = the frequency of the a allele. As an example, if p = 0.4 (40 percent) and q = 0.6 (60 percent), then p + q = 1, since the two alleles are the only ones present and the sum of the frequencies (or proportions) of all the alleles in a gene pool must equal 1 (or 100 percent). The Hardy-Weinberg principle states that at equilibrium the frequency of AA individuals will be p² (equal to 0.16 in this example), the frequency of Aa individuals will be 2pq, or 0.48, and the frequency of aa individuals will be q², or 0.36.

The basis of this equilibrium is that the individuals of one generation give rise to the next generation. Each diploid individual produces haploid gametes. An individual of genotype AA can make only a single type of gamete, carrying the A allele. Similarly, an individual of genotype aa can make only a gametes. An Aa individual, however, can make two types of gametes, A and a, with equal probability. Each individual makes an equal contribution of gametes, since all individuals are equally fit and there is random mating. Each AA individual will contribute twice as many A gametes as each Aa individual. Thus, to calculate the frequency of A gametes, add twice the number of AA individuals and the number of Aa individuals, then divide by twice the total number of individuals in the population (note that this is the same as the method to calculate allele frequencies). That means that the frequency of A gametes is equal to the frequency of A alleles in the gene pool of the parents.

The next generation is formed by gametes pairing at random (independent of the allele they carry). The likelihood of an egg joining with a sperm is the frequency of one multiplied by the frequency of the other. AA individuals are formed when an A sperm joins an A egg; the likelihood of this occurrence is p × p = p² (that is, 0.4 × 0.4 = 0.16 in the first example). In the same fashion, the likelihood of forming an aa individual is q² = 0.36. The likelihood of an A egg joining an a sperm is pq, as is the likelihood of an a egg joining an A sperm; therefore, the total likelihood of forming an Aa individual is 2pq = 0.48. If one now calculates the allele frequencies (and hence the frequencies of the gamete types) for this generation, they are the same as before: The frequency of the A allele is p = (2p² + 2pq)/2 (in the example, (0.32 + 0.48)/2 = 0.4), and the frequency of the a allele is q = (1 – p) = 0.6. The population remains at equilibrium, and neither allele nor genotype frequencies change from one generation to the next.

Ideal Versus Real Conditions

The Hardy-Weinberg equilibrium is a mathematical model of the behavior of ideal organisms in an ideal world. The real world, however, does not approximate these conditions very well. It is important to examine each of the five assumptions made in the model to understand their consequences and how closely they approximate the real world.

The first assumption is infinitely large population size, which can never be true in the real world, as all real populations are finite. In a small population, chance effects on mating success over many generations can alter allele frequencies. This effect is called genetic drift. If the number of breeding adults is small enough, some genotypes will not get a chance to mate with one another, even if mate choice does not depend on genotype. As a result, the genotype ratios of the offspring would be different from the parents. In this case, however, the gene pool of the next generation is determined by those genotypes, and the change in allele frequencies is perpetuated. If it goes on long enough, it is likely that some alleles will be lost from the population, since a rare allele has a greater chance of not being included. Once an allele is lost, it cannot be regained. How long this process takes is a function of population size. In general, the number of generations it would take to lose an allele by drift is about equal to the number of individuals in the population. Many natural populations are quite large (thousands of individuals), so that the effects of drift are not significant. Some populations, however, especially of endangered species, are very small.

The second assumption is that there is no differential migration, or movement of genotypes into or out of the population. Individuals that leave a population do not contribute to the next generation. If one genotype leaves more frequently than another, the allele frequencies will not equal those of the previous generation. If incoming individuals come from a population with different allele frequencies, they also alter the allele frequencies of the gene pool.

The third assumption concerns mutations. A mutation is a change in the DNA sequence of a gene—that is, the creation of a new allele. This process occurs in all natural populations, but new mutations for a particular gene occur in about one of 10,000 to 100,000 individuals per generation. Therefore, mutations do not, in themselves, play much part in determining allele or genotype frequencies. Yet, mutation is the ultimate source of all alleles and provides the variability on which evolution depends.

The fourth assumption is that there is random mating among all genotypes. This condition may be true for some genes and not for others in the same population. Another common limitation on random mating is inbreeding, the tendency to mate with a relative. Many organisms, especially those with limited ability to move, mate with nearby individuals, which are often relatives. Such individuals tend to share alleles more often than the population at large.

The final assumption is that all genotypes are equally fit. Considerable debate has focused on the question of whether two alleles or genotypes are ever equally fit. Many alleles do confer differences in fitness; it is through these variations in fitness that natural selection operates. Yet, newer techniques of molecular biology have revealed many differences in DNA sequences that appear to have no discernible effects on fitness.

Theoretical and Experimental Genetic Studies

The field of population genetics uses the Hardy-Weinberg equations as a starting place, to investigate the genetic basis of evolutionary change. These studies have taken two major pathways: theoretical studies, using ever more sophisticated mathematical expressions of the behavior of model genes in model populations, and experimental investigations, in which the pattern of allele and genotype frequencies in real or laboratory populations is compared to the predictions of the mathematical models.

Theoretical population genetics studies have systematically explored the significance of each of the assumptions of the Hardy-Weinberg equilibrium. Mathematical models allow one to work out with precision the behavior of a simple, well-characterized system. In this way, it has been possible to estimate the effects of population size or genetic drift, various patterns of migration, differing mutation rates, inbreeding or other patterns of nonrandom mating, and many different patterns of natural selection on allele or genotype frequencies. As the models become more complex, and more closely approximate reality, the mathematics becomes more and more difficult. This field has been greatly influenced by ideas and tools originally devised for the study of theoretical physics, notably statistical mechanics. Some of the most influential workers in this field were trained as mathematicians and view the field as a branch of applied mathematics, rather than biology. Consequently, many of the results are not easily understood by the average biologist.

Experimental population genetics tests predictions from theory and uses the results to explain patterns observed in nature. The major advances in this field have been determined, in part, by some critical advances in methodology. To study the behavior of genes in populations, one must be able to determine the genotype of each individual. The pattern of bands on the giant chromosomes found in the salivary glands of flies such as Drosophila form easily observed markers for groups of genes. Since these animals can be easily manipulated in the laboratory, as well as collected in the field, they have been the subjects of much experimental work. Using population cages, one can artificially control the population size, amount of migration, mating system, and even the selection of genotypes, and then observe how the population responds over many generations. More recently, the techniques of allozyme or isozyme electrophoresis and various methods of examining DNA sequences directly have made it possible to determine the genotype of nearly any organism for a wide variety of different genes. Armed with these tools, scientists can directly address many of the predictions from mathematical models. In any study of the genetics of a population, one of the first questions addressed is whether the population is at Hardy-Weinberg equilibrium. The nature and degree of deviation often offer a clue to the evolutionary forces that may be acting on it.

Understanding Genotypes

As the cornerstone of population genetics, the Hardy-Weinberg principle pervades evolutionary thinking. The advent of techniques to examine genetic variation in natural populations has been responsible for a great resurgence of interest in evolutionary questions. One can now directly test many of the central aspects of evolutionary theory. In some cases, notably the discovery of the large amount of genetic variation in most natural populations, evolutionary biologists have been forced to reassess the significance of natural selection compared with other forces for evolutionary change.

In addition to the great theoretical significance of this mathematical model and its extensions, there are several areas in which it has been of practical use. An area in which a knowledge of population genetics is important is agriculture, in which a relatively small number of individuals are used for breeding. In fact, much of the early interest in the study of population genetics came from the need to understand the effects of inbreeding on agricultural organisms. A related example, and one of increasing concern, is the genetic status of endangered species. Such species have small populations and often exhibit a significant loss of the genetic variation that they need to adapt to a changing environment. Efforts to rescue such species, especially by breeding programs in zoos, are often hampered by an incomplete consideration of the population genetics of small populations. A third example of a practical application of population genetics is in the management of natural resources such as fisheries. Decisions about fishing limits depend on a knowledge of the extent of local populations. Patterns of allele frequencies are often the best indicator of population structure. Population genetics, by combining Mendel’s laws with the concepts of population biology, gives an appreciation of the various forces that shape the evolution of the earth’s inhabitants.

In another application of the Hardy-Weinberg law, medical geneticists may evaluate the genes of two individuals or two animals to determine the likelihood of them producing offspring with harmful genetic mutations. These predictions can guide medical research in studying genes that cause certain diseases and conditions, like sickle-cell anemia and cystic fibrosis, that are most applicable to the population at a specific time. They can also help scientists breed specific traits into or out of groups to preserve a species. It can also be applied to ecological sciences. As particular environmental pollutants are scientifically linked to certain illnesses, the Hardy-Weinberg law can help predict who will be impacted and perhaps what genetic changes exposure may cause.

Principal Terms

Allele: One of several alternate forms of a gene; the deoxyribonucleic acid (DNA) of a gene may exist as two or more slightly different sequences, which may result in distinct characteristics

Allele Frequency: The relative abundance of an allele in a population

Diploid: Having two chromosomes of each type

Gene: A section of the DNA of a chromosome, which contains the instructions that control some characteristic of an organism

Gene Pool: The array of alleles for a gene available in a population; it is usually described in terms of allele or genotype frequencies

Genotype: The set of alleles an individual has for a particular gene

Genotype Frequency: The relative abundance of a genotype in a population

Haploid: Having one chromosome of each type

Population: The individuals of a species that live in one place and are able to interbreed

Random Mating: The assumption that any two individuals in a population are equally likely to mate, independent of the genotype of either individual

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