Hardy-Weinberg Equilibrium and the Modes of Evolution

Population Genetics, Hardy-Weinberg Equilibrium, and the Modes of Evolution

NOTE: These are lecture notes for Biology 391, Organic Evolution, at The University of Tennesee at Martin. Anyone outside of UT Martin wishing to use these notes or to contact me for additional information should first read the information obtained by clicking here.

Goals: the goals of this lecture are to introduce the study of population genetics and to identify what we expect in a population when no evolution is occurring (a situation called Hardy-Weinberg Equilibrium.) This will allow us to identify the situations in which a population is NOT in Hardy-Weinberg Equilibrium; these are situations in which evolution does occur.

Related Textbook Material: Freeman and Herron (2001) Chapter 5

Lab Manual Questions over this material are in Lab Manual Chapter IV

The Lecture:

Population genetics refers to the study of evolution that is done by determining or modeling genetic changes in populations.

. There are two main measures of the genetics of a population:

allele frequency: the proportion of a specific allele out of all the alleles in the all the individuals in a population
genotype frequency: the proportion of a specific genotype out of all the genotypes in all the individuals in a population

To understand what these mean, consider a population made of N individual organisms. Suppose that they are diploid, and reproduce sexually. Consider one gene with two alleles, A and a. The possible genotypes are therefore AA, Aa, and aa.

First lets define the allele frequencies mathematically. We could determine th e allele frequency of allele A and the allele frequency of allele a. To do this, we would count all the alleles in the population, and determine the proportion of them that are A and the proportion of them that are a. So, if #A means the number of A alleles counted up from all individuals in the p opulation, #a means the number of a alleles counted up from all individuals in the population, and N is the number of individuals in the population, then:

Note that #A + #a = 2N because each individual has two alleles, so the total number of alleles in the population (#A+#a) is twice the number of individuals in the population.

Note also that allele frequencies are proportions. Since all proportion a dd up to the whole (to one), the allele frequencies must add up to one. That is:

Now let's define the genotype frequencies mathematically. We could determine genotype frequencies for AA, Aa and aa by counting up all individuals with each genotype in the population and then determining the proportion of all the individuals that are AA, the proportion that are Aa, and the proportion that are aa. So, if #AA is the number of AA individuals, #Aa is the number of Aa individuals, and #aa is the number of aa individuals, the genotype frequencies are:

Note that #AA +#Aa+#aa = N because each individual has one genotype, and there are N individuals in the population, so adding up all the genotypes in the population (by adding #AA+#Aa+#aa) gives the total number of individuals in the population, which is N.

The formulas given above are simply definitions of allele and genotype frequencies. You need to study them to understand what allele and genotype frequencies mean, but you won't use them much to solve problems. To study evolution, we w ill use these definitions to develop relationships between allele and genotype frequencies, and how they change over time, when evolution is occurring and when it is not occurring. We will develop the formulas you do need to use to solve problems in the n ext section, and the next lecture.

Now that we've defined allele and genotype frequencies, we can give a definition of evolution in terms of those frequencies. Genetically we can define:

evolution is change in allele and/or genotype frequency over time.

To be able to see if evolution is occurring, we need to consider what we would expect if evolution is NOT occurring. Once we know that, then if we see anything different from that we know we are observing evolution.

Hardy-Weinberg Equilibrium is defined as the situation in which no evolution is occurring.

A trait (coded for by some gene with alternate alleles) in a population will be in Hardy-Weinberg Equilibrium if five assumptions are met. These are:

infinite population size (there are infinitely many individuals in the population)
there is no movement of individuals from population to population
there is no mutation (no biochemical changes in DNA that produce new alleles.)
there is random mating (this means that with regard to the trait we're looking at, individuals mate at random -- they don't select mates based on this trait in any way.)
the different genotypes (for the genetic trait we're studying) have equal fitness. This means there is no natural selection affecting this trait (remember one of the conditions for natural selection is that some traits result in higher fitness than others.)

We will now see how allele frequencies and genotype frequencies are related, and show that evolution does not occur, in Hardy-Weinberg Equilibrium. To do this, we are going to follow allele and genotype frequencies through a generation, starting with the gametes (eggs and sperm) that will unite to make the individuals of a generation, and finishing with the gametes that these individuals make to start the next generation.

We are going to continue to consider a population of diploid, sexually reproducing individuals, and we will look at a gene with two alleles, A and a, so there are three genotypes, AA, Aa, and aa.

In the gametes that start a generations, call the frequency of A "p" and the frequency of a "q" (that is, Freq(A) = p and Freq(a)=q.)

We will also assume this population meets the five assumptions of Hardy-Weinberg Equilibrium that are listed above.

Because mating is random, alleles mix together at random to give genotypes of zygotes (fertilized eggs.)

Because the population is infinitely large, the probability of getting a gamete with a particular allele in it is simply the frequency of that allele. Similarly, determining the probabilities of getting particular genotypes will tell us the frequencies of those genotypes in the population.

So to get the homozygous genotype AA, you get an egg with allele A, with probability p, and a sperm with allele A, also with probability p. The probability of getting both of these is determined by multiplying the probabilities of each event together. That is:

The genotype frequency of AA will be (p)(p), or p²

Similarly, for the other homozygote, genotype aa, the chance of a sperm with an "a" allele is q and the chance of an egg with an "a" allele is also q, so:

The genotype frequency of aa will be (q)(q), or q²

There are two ways of getting the heterozygous genotype, Aa. These are:

an "A" bearing sperm and an "a" bearing egg
an "A" bearing egg and an "a" bearing sperm

The probability of an "A" bearing sperm is p; the probability of an "a" bearing egg is q, so the probability of the first way of getting an Aa is (p)(q) = pq.

Similarly, the probability of an "A" bearing egg is p; the probability of an "a" bearing sperm is q, so the probability of the second way of getting an Aa is also (p)(q)=pq.

So to get the probability of getting the Aa genotype, we add together the probabilities of the two ways of getting this genotype. So:

The genotype frequency of Aa is pq+pq=2pq

NOTE: when a population is in Hardy-Weinberg Equilibrium, there is a mathematical relationship between the allele frequencies and the genotype frequencies. This means that if you know allele frequencies, you can solve for genotype freq uencies, and if you know genotype frequencies, you can solve for allele frequencies. YOU WILL NEED TO BE ABLE TO DO THIS! A number of the practice questions in Chapter VI of your lab manual will allow you to practice this. To solve these, you will need to remember that if the allele frequencies are p and q, then the genotype frequencies are p², 2pq, and q². You should also remember that allele frequencies must always add up to one, so p+q=1, and that genotype frequencies must always add up to one, so p²+2pq+q²=1.

Now let's see what happens to these genotype and allele frequencies over time. We have said that no evolution is occurring, so they should not change. We can show that this is true.

Since genotypes have equal fitness, there is nothing to cause them to change, so the adults that develop from these zygotes will have same genotype frequencies. These adults will then produce gametes to start the next generation. The allele frequencies in these gametes based on the frequencies of adults that produce each type of gamete, as follows.

Since all gametes from AA adults will have allele A, and half the gametes from Aa adults will have allele A, the frequency of A that is in the adults of the population, and that will be passed on to the next generation, must be:

We have determined that Freq(AA)=p² and that Freq(Aa)=2pq. Substituting these values into the above equation, we find that:

Freq(A)= p²+(1/2)2pq = p²+pq

To figure out what this equals, we first factor out p:

Freq(A)= p²+pq = p(p+q) and since p+q=1, this equals p.

So the frequency of allele A has not changed. Since p+q=1, if p hasn't changed, then q, the frequency of a, has not changed, either.

Since the genotype frequencies in the next generation will be based on these allele frequencies, just as they were in this generation, genotype frequencies won't change from generation to generation either.

What this means is that in Hardy-Weinberg Equilibrium, allele frequencies will stay the same from generation to generation to generation. Genotype frequencies will also stay the same. This is interesting because it means that sexual re production, by itself, does not change genotype or allele frequencies. Any time the assumptions of Hardy-Weinberg Equilibrium are met, there is no genetic change -- evolution does not occur.

A major way in which understanding Hardy-Weinberg Equilibrium is important to understand evolution is that the five assumptions of Hardy-Weinberg Equilibrium must all be met for there to be no evolution. When any one of them is not met, evolution occur . We can define five different ways in which evolution occurs based on the situations in which these five assumptions are NOT met. These are:

genetic drift is change in allele frequency by random chance. It occurs if a population is not infinite in size. In populations that are not infinitely large, there will be random error in which alleles are passed from generation, and allele frequencies will change at random. Since no population is really infinitely large, there is always some genetic drift occurring; however, the effect is very small in large populations. The effect of genetic drift is larger and larger in smaller an d smaller populations.
gene flow is change in allele frequency that occurs because individuals move among populations. If there are different allele frequencies in different populations of a species, then when individuals move to a new population, they will add alleles in proportions different from those in the population, and therefore change the allele frequencies in the population.
mutation is biochemical change in DNA. It changes one allele into another, and can create new alleles. It is a very unusual process (typical mutation rates are about one mutation in a million genes passed from generation to generation ); as a result, evolution through mutation is extremely slow -- so slow we can't generally detect it. However, mutation is important for evolution because, ultimately, mutation is the source of genetic variation and other forms of evoluti on can not occur without genetic variation. So without mutation, the other, faster forms of evolution would not occur, either. Another important point about mutation is that, with regard to the fitness of alleles, it is random -- it may produce alleles th at result in high fitness or low fitness. What happens to those alleles, once they are produced, depends on the other forms of evolution such as natural selection.
non-random mating is evolution that occurs because individuals select mates based on their characteristics. Some forms of non-random mating change genotype, but not allele frequencies; others change both. (Note that your textbook uses a somewhat restricted definition of non-random mating as only changing genotype frequencies and note that the authors do not consider this evolution. We will take a broader view and consider -- later in the course -- any situation in which individuals ar e choosing mates or in which a trait is affecting mating success to be an example of non-random mating, and there are many situations in which this changes both allele and genotype frequencies.)
natural selection is evolution that occurs because different genotypes have different fitness. We have already defined and discussed natural selection based on different traits. This definition is a little more specific since it notes that it is different genotypes that have different fitness, but it is saying basically the same thing. You should review the lecture on natural selection and make sure you understand that this is the same before proceeding to the next lecture. In the nex t lecture, we will develop a mathematical model of natural selection, and it is crucial that you understand natural selection non-mathematically first so that you will be able to translate it into mathematics!

Study Tips for doing Hardy-Weinberg Equilibrium Problems:

make sure you know what your starting point is. To start a problem, you could be given allele frequencies, genotype frequencies, or phenotype frequencies that would allow you to determine genotype frequencies.
if you are given one allele frequency (p, for example) you can determine q since p+q=1
if you are given allele frequencies at the start of the problem, you can use them to solve for the genotype frequencies as p2, 2pq, and q2
if you are given genotype (or phenotype) frequencies at the start of the problem, you need to get allele frequencies to complete the problem. Remember you can get the allele frequency of an allele by taking the square root of the homozygote for that allele (for example, the allele frequency of D would be the square root of the genotype frequency of genotype DD.) Once you have the allele frequencies, you can solve for any other genotype frequencies.

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