Studies in population genetics
(modified from: Morgan, J.G., and M.E.B. Carter, Investigating Biology (4th edition), San
Francisco: Benjamin Cummings, 2002.
Objectives:
1. To explain the Hardy-Weinberg equilibrium in terms of allelic and genotypic frequencies and
relate these to the expression (p + q)2= p2 + 2pq + q2 = 1.
2. To describe the conditions necessary to maintain this equilibrium
3. To use the bead model to demonstrate conditions for evolution.
Background:
Charles Darwin proposed a mechanism for evolutionary change- natural selection, the differential
survival and reproduction of individuals in a population. In On the Origin of Species (1859),
Darwin described natural selection and provided abundant evidence in support of evolution, the
change in populations over time. Evolution was accepted as a theory with great explanatory
power supported by a substantial body of evidence. However, at the turn of the century,
geneticists and naturalists disagreed about the role of natural selection and the importance of
small variations in populations. How could these variations provide a selective advantage that
would result in evolutionary change? It was not until evolution and genetics became incorporated
into population genetics that natural selection became widely accepted.
A population is defined as a group of organisms of the same species that share a common gene
pool, all the alleles at all gene loci of all individuals in the population. The population is
considered the basic unit of evolution. Populations evolve, not individuals. How is this explained
in terms of natural selection?
In 1908, English mathematician G.H. Hardy and German physician W. Weinberg independently
developed models of population genetics that showed that the process of heredity itself did not
affect the genetic structure of a population. The Hardy-Weinberg theorem states that the
frequency of alleles in the population will remain the same regardless of the starting frequencies.
The equilibrium genotypic frequencies will be established after one generation of random mating.
This theorem applies only if:
1. The population is very large;
2. Matings are random;
3. There is no net change in the gene pool due to mutation;
4. There is no migration of individuals into or out of the population;
5. There is no selection; all genotypes are equal in reproductive success.
Basically, the Hardy-Weinberg theorem provides a baseline model in which gene frequencies do
not change and evolution does not occur. By testing the fundamental hypothesis of the HardyWeinberg theorem, evolutionists have investigated the roles of mutation, migration, population
size, nonrandom mating, and natural selection in effecting evolutionary change in natural
populations. Although some populations maintain genetic equilibrium, the exceptions are
intriguing.
Use of the Hardy-Weinberg theorem
The theorem provides a mathematical formula for calculating the frequencies of alleles and
genotypes in populations. If we begin with a population with two alleles at a single gene locus- a
dominant allele, A, and a recessive allele, a- then the frequency of the dominant allele is p, and
the frequency of the recessive allele is q. Therefore, p + q =1. If the frequency of one allele, p, is
known, then the frequency of the other allele, q, can be determined by using the formula q = 1- p.
During sexual reproduction, the frequency of each type of gametes produced is equal to the
frequency of the alleles in the population. If the gametes combine at random, then the probability
of AA in the next generation is p2, and the probability of aa is q2. The heterozygote can be
obtained two ways, with either parent providing the dominant allele, so the probability is 2pq.
These genotypic frequencies can be obtained by multiplying p+q by p+q. This can also be
expressed as p2 + 2pq + q2 =1.
To summarize:
p2 = frequency of AA
2pq = frequency of Aa
q2 = frequency of aa
Example:
If alternative alleles of a gene, A and a, occur at equal frequencies, p and q, then during sexual
reproduction, half (0.5) of all gametes will carry A and half will carry a.
What is the frequency of p? _________ What is the frequency of q? ___________
Recall the formula for calculating the genotypic frequencies with respect to p and q.
What is the predicted frequency of AA? ______ of Aa? _______ of aa? ________
Do the allelic frequencies change? __________
In actual populations the frequencies of the alleles are NOT usually equal. Recessive traits are
often rare. For example, 4% of the populations might be albino, which is a recessive trait. The
frequency of the albino genotype (homozygous recessive, or q2) is 0.04. What is the frequency of
the albino ALLELE?
What is the frequency of the allele for normal pigmentation? ______________
Under Hardy-Weinberg conditions, we have now established the equilibrium frequencies of the p
and q alleles. Thus, in each succeeding generation, what is the frequency of the following
genotypes:
AA? _________ Aa? __________ aa? ___________
This equilibrium will continue indefinitely if the conditions of the Hardy-Weinberg theory are
met. Although this generally does not occur in nature, this equilibrium provides a valuable model
from which we can predict genetic changes in populations as a result of natural selection or other
factors. This allows a quantitative approach to the study of evolution at the population level.
PART I. Testing the Hardy-Weinberg equilibrium using a bead model
Materials
Bag containing 100 beads of two colors
Introduction
Work in pairs. You will simulate a population using a collection of colored beads, which
represents the gene pool of a population. Each bead should be regarded as a single gamete, and
the two colors represent the two alleles of a single gene. Each bag will contain 100 beads, in
preset proportions. Record in the space below the colors of the beads and the initial frequencies
for your gene pool.
A= ________________ color ______________ allelic frequency
a= ________________color _______________ allelic frequency
1. How many diploid individuals does this represent? ____________
2. What colors of beads are seen in a homozygous dominant individual? __________
3. What colors of beads are seen in a homozygous recessive individual? __________
4. What would be seen in a heterozygous individual? _________________
Hypothesis
State the Hardy-Weinberg theory in your own words.
Prediction:
Predict the genotypic frequencies of the population in future generations.
Procedure:
1. Without looking, remove two beads from the bag. These two beads represent one
individual in the next generation. Record the diploid genotype (AA, Aa or aa) of the individual
thus formed.
2. Return the beads to the bag and shake to mix the beads. By replacing the beads each
time, the size of the gene pool remains constant and the probability of selecting any allele should
remain equal to its frequency. This technique is called sampling with replacement.
3. Repeat steps 1 and 2, recording each result, until you have recorded genotypes for 50
individuals who will form the next generation of the population.
Results:
1. First, calculate the expected frequencies of frequencies of genotypes and alleles for the
population. Use the original allelic frequencies for the population that was provided. Record the
numbers in the spaces provided (the top set).
Parent populations
New populations
Allelic frequency
A
a
Genotypic number and frequency
AA
Aa
aa
______
Allelic frequency
A
a
______
______
_______
______
______
______
______ _______
______
______
______
______
______
2. Next, calculate the observed frequencies, using the genotypes of your 50 individuals. Fill in the
results in the second set of spaces provided.
Discussion
Compare your results with those of other teams. How variable are the results for other teams?
What would you expect to happen if you continued the simulation for 25 more generations?
Is this population evolving? Why or why not?
Simulation of Evolutionary Change Using the Bead Model
Under the conditions specified by the Hardy-Weinberg model, the genetic frequencies should not
change, and evolution should not occur. In this exercise, we will modify the conditions and
determine the effect of genetic frequencies on future generations. We will simulate the changes
that occur when the Hardy-Weinberg conditions are not met.
You will simulate two of the experimental scenarios and, using the bead model, determine the
changes in genetic frequencies over several generations. These include the migration of
individuals between two populations, also called gene flow; the effect of small population size,
called genetic drift; and examples of natural selection.
The experiments will be carried out with the procedures described below.
1. Sampling with replacement
The gene pool size will be 100 beads. The frequency of each allele (i.e., the number of beads
representing A and the number representing a, will be preset.) Each new generation will be
formed by randomly choosing 50 diploid individuals formed by pairs of beads. After removing
each pair of beads (and recording the genotype) Replace the pair, shake the bag of beads to mix,
and THEN remove the next set (sampling with replacement).
2. Reestablishing a population with new genetic frequencies
In some cases, the number of individuals will decrease as a result of the simulation. In those
cases, return the population to 100 beads but with the new allelic frequencies. For example, if you
form a population as previously described and eliminate all homozygous recessive individuals
(aa) by selection, then the resulting frequencies would be:
Number of individuals: 14 AA, 24 Aa, 0 aa
Number of beads (alleles) 28 A + 24 A (=52); 24 a
Total number of alleles: 76
Frequency of A: 52/76 = 0.68
Frequency of a: 24/76 = 0.32
To reestablish a population of 100 alleles, add beads in the new proportions: 68 beads
representing the A allele, and 32 representing the a allele.
Part 1: simulation of genetic drift
Introduction
Genetic drift is the change in allelic frequencies in a small population due to chance alone.
Combinations of gametes may not be random, due to sampling error. If you toss a coin 500 times,
you will probably get a 50:50 ratio of heads: tails. If you toss the coin only 10 times, the ratio
may deviate substantially by chance alone. Genetic fixation, the loss of all but one possible allele
at a gene locus in a population, is a common result of genetic drift in small natural populations.
Genetic drift is a significant evolutionary force in situations known as the bottleneck effect and
the founder effect.
Bottleneck Effect
The bottleneck effect occurs when a population undergoes a drastic reduction in size as a result of
chance events, such as natural disasters. This is often simulated as beads passing through a
bottleneck, which results in an unpredictable combination of beads that pass through. These beads
(alleles) constitute the beginning of the next generation.
Procedure
1. Establish a starting population of 50 individuals with a frequency of 0.5 for each allele
(how many beads of each color?). Thus is generation 0.
2. Without replacement, randomly select 5 individuals (how many pairs of alleles. Record
the genotypes of these individuals. Determine the genotypic frequencies of AA, Aa and
aa and determine the new allelic frequencies, p and q. Record these values in the table.
(observed frequencies)
3. Using the new observed frequencies, calculate the expected genotypic frequencies (p2,
2pq, q2) and record these values under generation 1, observed frequencies.
4. Reestablish the population to 50 individuals using the new allelic frequencies. Repeat
steps 2 and 3. Record the calculations and observed results for each generation.
5. Keep repeating until one of the alleles becomes fixed in the population for at least two
generations. (Be sure to establish each generation with the new frequencies.)
How many generations were simulated?
Graph the change in p and q over time. Did one allele go to fixation? If so, which one? How do
your results compare with others?
Generation AA
Aa
aa
A (p)
a (q)
p2
2pq
q2
observed Observed observed observed observed expected expected expected
0
0.5
0.5
0.25
0.50
0.25
1
2
3
4
5
6
7
8
9
10
Analysis:
Do you see a consistent trend in allelic frequency and change? Why or why not?
If you were to repeat the experiment 100 times, starting with allelic frequencies of p=q=0.5, how
frequently would you expect allele A to go to fixation? Allele q? What if the allelic frequencies
were p=0.2 and q=0.8?
Other simulations- abbreviated instructions are given, since the procedures have already been
described in detail. You will be assigned one of these, in addition to the bottleneck effect
simulation. We will discuss your results as a group upon completion.
Founder effect
When a small group of individuals becomes separated from the larger parent population, the
allelic frequencies in this small gene pool may be different from those of the original population
by chance alone. This occurs when a group of migrants becomes established in a new area- such
as the colonization of an island1. Establish a population of 50 individuals with your choice of allelic frequencies. Record
this information in the table provided. (generation 0)
2. Without replacement, randomly select 5 individuals and record their genotypes and allelic
frequencies. This is the founder population. Calculate the new allelic and genotypic
frequencies for this population.
3. Reestablish the population to 50 individuals using the new allelic frequencies. Each
generation is produced by sampling with replacement. Continue for several generations,
establishing the population with the obtained allelic frequency each time.
4. Summarize the results. Compare the founder population with the original population. Is
the equilibrium frequency different? What results were obtained by other groups?
Gene flow: simulation of migration
The migration of individuals between populations results in gene flow. In a natural population,
this can result from immigration and emigration of individuals or gametes (e.g., through pollen
movement). The rate and direction of migration and starting allelic frequencies can affect the rate
of genetic change. In this case, the migration rate is the same but the starting allelic frequencies
are different.
1. This simulation requires two groups with populations of 50 diploid individuals, with
different starting allelic frequencies. Record the frequencies in the Generation 0.
2. Select 10 individuals from each population and allow them to migrate to the other
population by moving the beads. (Both groups will now have 100 beads.
3. Select 50 individuals, sampling with replacement. Record the genotypes, etc. and
establish the population with the new allelic frequencies.
4. Repeat steps 2 and 3 for several generations, until you can draw some sort of conclusion.
Simulation of natural selection
Natural selection, the differential survival and reproduction of individuals, was first proposed by
Darwin as the mechanism for evolution. (As the previous simulations attest, natural selection is
important, but not the only contributor to evolution.) Natural selection is based on the observation
that individuals with certain heritable traits are more likely to survive and reproduce than those
lacking these advantageous traits. Therefore, the proportion of offspring with these traits will
increase in the next generation. Whether traits are advantageous in a population depends on the
environment and the selective agents (physical and biological). We will model natural selection if
the allele for sickle-cell trait.
Sickle-cell disease is caused by a mutant allele (h) which, in the homozygous condition (hh), was
often fatal at a young age. The mutation causes the formation of abnormal, sickle-shaped red
blood cells that clog blood vessels, cause organ damage, and transport oxygen very poorly.
Individuals who are heterozygous (Hh) have the sickle cell trait, which is not fatal and is often
asymptomatic. Scientists were initially surprised at the high frequency of the allele in Africa until
they realized that heterozygous individuals have a selective advantage in resisting malaria. Thus,
although the homozygous condition is often fatal, there is strong pressure to maintain the allele.
1. Establish a population of 50 individuals with allelic frequencies of 0.5 for each allele. Do
what you usually do for generation 0.
2. Assume the selective force for each genotype as follows:
a. HH homozygotes: 30% die of malaria
b. Hh heterozygotes: 10% die of malaria
c. hh homozygotes: 100% die of sickle-cell disease
3. Determine the genotypes and phenotypes of the population by selecting 50 individuals,
sampling with replacement. Eliminate individuals according to the per cent mortality
shown above.
4. Calculate the new allelic and genotypic frequencies based on the survivors, and
reestablish the population using these new frequencies. Continue the process for several
generations.
Generation AA
Aa
aa
A (p)
a (q)
p2
2pq
q2
observed Observed observed observed observed expected expected expected
0
0.5
0.5
0.25
0.50
0.25
1
2
3
4
5
6
7
8
9
10