Introduction

The Hardy-Weinberg Theorem is a mathematical formulation that allows you to interrelate allele frequencies and genotype frequencies in a population of diploid or polyploid individuals. Alleles are different forms of a particular gene. There can be one, two or many alleles for a particular gene. In a population of diploid individuals, each individual carries two copies of a particular gene, and the copies may be the same or different alleles. The genetic composition of each individual is its genotype, a term which refers to which alleles the individual carries, and whether the individual is homozygous or heterozygous. In the population as a whole, we can add up the total number of different alleles present. Each diploid individual carries two copies of a gene, so the total number of copies of that gene in the population is 2X the number of individuals in the population. Of this total number of copies, some will consist of one allele, others will consist of another allele, and so on. The composition of alleles in the population is its gene pool, and we can specify the fraction of the gene pool consisting of different alleles. For a gene that has only two alleles, we might specify the frequency of one allele as p (a fraction between 0 and 1), and the frequency of the other allele as q. The sum of p and q will be 1.0. These are the allele frequencies in the population. We can also count up the frequencies of different genotypes in the population. If there are only two alleles, we would have homozygotes for one allele, homozygotes for the other allele, and heterozygotes; the fraction of individuals in the population with each of these genototypes is the genotype frequency. Again, the sum of the different genotype frequencies will be 1.0.

The Hardy-Weinberg Theorem relates the frequencies of alleles and genotypes in the following way. If p and q are the frequencies of the only two alleles (for a particular gene) in the population, the genotype frequencies can be expressed as:

where p² is the frequency of organisms homozygous for the first allele, q² is the frequency of organisms homozygous for the alternate allele, and 2pq is the frequency of heterozygous organisms. Using this formula, you could determine the allele frequencies in a population if you knew the genotype frequencies, and the genotype frequencies if you knew the allele frequencies.

The Hardy-Weinberg Theorem applies only if no forces are operating to change the frequency of the alleles in the population. If that is true, the frequency of the various genotypes will also remain unchanged from generation to generation. Forces that may (and almost always do) operate to change allele frequencies include:

In addition, non-random mating, while having no effect on allele frequencies, may result in genotypes frequencies different from those predicted by the Hardy-Weinberg Theorem.

But in the absence of any of these factors the allele frequencies and the frequencies of the different genotypes will remain the same from one generation to the next. Genetic recombination due to sexual reproduction will not of itself result in any changes in allele frequencies or genotype frequencies.

Therefore, if we observe a population for a few generations and notice that allele frequencies and/or genotype frequencies for a particular trait change from generation to generation, we conclude that at least one of these factors - mutation, migration, natural selection, chance effects or non-random mating - is operating on the trait in the population we are studying.

In nature, most populations do in fact exhibit changing allele and genotype frequencies, for the simple reason that in most natural populations factors such as those mentioned above are almost always at work. Of what possible use, then, is the Hardy-Weinberg Theorem, if it describes a situation that we almost never encounter in nature? The answer is that it provides a standard that we can use to assess the changes that we observe taking place in the genetic make-up of populations.

Peppered Moths

In England during the 19th century populations of peppered moths (Biston betularia) changed dramatically. The proportion of dark individuals in those populations increased, and the proportion of light individuals declined. This change was especially striking in the heavily industrialized areas of the country. The Hardy-Weinberg Law enables us to see that a change such as this one in the genetic composition of populations could not have taken place as the result of genetic recombination through sexual reproduction alone. Instead, we must look for other factors which might account for this change. In the case of the peppered moths, most evolutionary biologists now agree that changing selective pressures on these populations as a result of industrialization caused the numbers of dark individuals to increase and the numbers of light individuals to decline.

In a sense the Hardy-Weinberg Law can be compared to Newton's First Law of Motion, which states that a body at rest tends to remain at rest, and a body in motion tends to remain in motion (and in motion of the same speed and direction), unless the body is acted upon by some force. Now in nature it would probably be impossible to find a body of any kind that was not being acted upon by some force - usually by several. But this does not mean that Newton's First Law is wrong. Instead, it means that we can look at real bodies in nature and at the ways in which their motion is changing and use Newton's First Law to deduce the forces which must be acting on them. In the same way, we can look at the way the genetic composition of natural populations changes, and use the Hardy-Weinberg Law to help us to identify the nature and gauge the magnitude of the factors that are operating to effect those changes.

Where does the Hardy-Weinberg equation come from?

Where does the Hardy-Weinberg equation come from, how do we know it's true, and what is its biological meaning? To begin with, it is possible to offer a simple algebraic proof of the equation if we remember that p represents the frequency of one allele, that q represents the frequency of another allele, and that for the gene in question there are only two alleles. This means that the frequency of the two alleles adds up to one. If we are told that all of the students in a class are either chemistry majors or biology majors, and that two tenths of them are chemistry majors, we know that eight tenths of them are biology majors and that ten tenths of them are one or the other. If p and q represent the two frequencies this fact can be expressed by the equation

Furthermore, if p + q = 1, then the quantity (p + q) raised to any power is also equal to one, because no matter how many times you multiply one by itself the product is always one. This can be expressed by the equation

If we take one particular case of this general expression and multiply it out, we end up with the Hardy-Weinberg equation in its familiar form:

We can derive the terms in the Hardy-Weinberg equation from simple probability theory. The probability of two independent events both occurring is equal to the product of their individual probabilities. If you flip a coin the probability of its being heads is 1/2 or 0.5. If you flip a second coin the probability that it will be heads is also 1/2 or 0.5. However, the probability that both coins will turn up heads is 1/2 x 1/2, or 1/4. If you flip three coins the probability that all three will turn up heads is 1/2 x 1/2 x 1/2, or 1/8. Remember that these are independent events - that one coin turns up one way has no effect on the outcome of other tosses.

Application to sexual reproduction and genotypes.

How does this apply to sexual reproduction and genotypes? Imagine that we're interested in a trait determined by a gene with two alleles, and these alleles exhibit Mendelian dominance. The dominant allele can be designated by a capital B and the recessive allele can be designated by a lower case b. Suppose further that in the population the frequency p of the dominant allele is 0.2 and that the frequency q of the recessive allele is 0.8. We're talking about a gene pool (more accurately an allele pool) in which 20% of the alleles are dominant B's and 80% are recessive b's.

Each newborn organism draws one allele from the maternal allele pool, and a second allele from the paternal allele pool. (Usually, allele frequencies are the same in the male and female parts of a population.) Given this situation, what is the chance that the organism will be homozygous for the dominant allele? The probability of drawing a B allele from the maternal pool is the frequency p of that allele, or 0.2. The probability of drawing a B from the paternal allele pool is also p, or 0.2. The probability of drawing a B from both of those pools is equal to the product of the two individual probabilities--p x p, or p², which in this case is equal to (0.2) x (0.2), or 0.04. Thus in this population 4% of the individuals are likely to have the genotype BB.

Similarly, the likelihood that a would-be organism will draw a recessive allele b from the maternal allele pool is q, or in this case 0.8. The likelihood of a recessive allele being drawn from the paternal allele pool is also 0.8, and the likelihood of drawing a recessive allele from both pools is q x q, or q², which in this case is (0.8) x (0.8), or 0.64. So in this population we would expect 64% of the individuals to be homozygous for the recessive allele (bb).

What is the probability that an organism produced from this population will be heterozygous for the trait? The probability of drawing a dominant allele B from the maternal pool is p, and the probability of drawing a recessive allele from the paternal pool is q, so the probability of ending up heterozygous in this way is equal to p x q, or pq, which in the case of our example is equal to (0.2) x (0.8), or 0.16. But this is not the only way in which a heterozygous organism can be produced. Such an organism also results if a recessive allele b is drawn from the maternal pool and a dominant allele B is drawn from the paternal pool. The probabilities are the same as in the first case, but since there are two ways in which a heterozygous organism can be produced, not just one, the expression for the frequency of heterozygotes in the population is 2pq, or in our case 2(0.2)(0.8), or 0.32. Therefore, 32% of the individuals in this population are likely to be heterozygotes.

The frequency of organisms homozygous for the dominant allele in our population is p², the frequency of organisms homozygous for the recessive allele is q², and the frequency of heterozygous organisms is 2pq. Since these are the only three possible genotypes, we know that the sum of their frequencies must be equal to one. If we were to write the preceding statement mathematically, the resulting expression would turn out to be none other than the now-familiar Hardy-Weinberg equation:

If we plug in numbers from our example, we can confirm the truth of the expression:

That, then, is the derivation and biological meaning of the Hardy-Weinberg equation.

Lab exercises:

• Part 1: Hardy-Weinberg Simulation
• Part 2: Simulations in Evolution using Populus
• Review Questions