Lab 6: Population Genetics: Hardy-Weinberg Theorem
| OBJECTIVES
After completing this exercise, you should be able to:
1) Explain Hardy‑Weinberg equilibrium in terms of allelic and genotypic frequencies and relate these to the expression (p + q)2 = p2 + 2pq + q2 = 1 .
2) Describe the conditions necessary to maintain Hardy‑Weinberg equilibrium.
3) Use a computer simulation to demonstrate conditions for evolution.
4) Test hypotheses concerning the effects of evolutionary change (migration, genetic drift via bottleneck effect, and natural selection) using a computer model.
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Simulation of Evolutionary Change Using an Online Population Simulator
Under the conditions specified by the Hardy‑Weinberg model (random mating, large population, no mutation, no migration, and no selection), the genetic frequencies should not change, and evolution should not occur. In this exercise, the class will modify these conditions and determine the effect on genetic frequencies in subsequent generations.
Work in teams of two or three students to simulate three scenarios:
1. Genetic drift- simulated by applying a bottleneck event where a population experiences a drastic reduction in size due only to chance (no selection)
2. Gene flow- simulated by allowing migration to occur between populations
3. Natural selection- simulated by providing certain genotypes with different levels of fitness.
You will be using a computerized population simulator, a general introduction is shown below. The URL for the simulator is http://www.radford.edu/~rsheehy/Gen_flash/popgen/
Experiment A. Simulation of Genetic Drift
Introduction
Genetic drift is the change in allelic frequencies in small populations as a result of chance alone. In a small population, combinations of gametes may not be random, owing to sampling error. (If you toss a coin 500 times, you expect about a 50:50 ratio of heads to tails; but if you toss the coin only 10 times, the ratio may deviate greatly in a small sample owing to 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 (investigated here) and the founder effect.
A bottleneck occurs when a population undergoes a drastic reduction in size as a result of chance events (not differential selection), such as a volcanic eruption, hurricane, or sometimes human influence. (Bad luck, not bad genes!) In Figure 6.1, the marbles pass through a bottleneck, which results in an unpredictable combination of marbles that pass to the other side. These marbles would constitute the beginning of the next generation, but the allelic frequencies might be entirely different than the original population! In the computerized simulation, both the size and duration (number of generations) of a bottleneck can be altered. Each variable can influencing how the allelic frequencies are impacted.
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Figure 6.1. The bottleneck effect. The gene pool can drift by chance when the population is drastically reduced by factors that act unselectively Bad luck, not bad genes! The resulting population will have unpredictable combinations of genes. What has happened to the amount of variation? |
Hypothesis
As your hypothesis, either propose a hypothesis that addresses the bottleneck effect specifically or state the Hardy-Weinberg theorem.
Prediction
Either predict equilibrium values as a result of Hardy-Weinberg or predict the type of change that you expect to occur in a small population (if/then). Deductive Reasoning
Procedure
1. Begin by going to the following webpage: http://www.radford.edu/~rsheehy/Gen_flash/popgen/ Then, setting the following parameters (in the image) on the computer simulation. You will be running the model five times, simulating five populations. The top graph of the output will represent the allelic frequencies, the bottom graph is displaying the genotype frequencies.
2. To investigate the bottleneck effect, select the “Bottle Neck” box and have the “Start” begin at generation 50 and the “End” at generation 100. Set the “BN Pop.” to 50 individuals. This represents a drastic reduction in population size over 50 generations. Fill in the information for this population in Table 6.1. An example is provided using the graph below.
3. Click the “GO” button to run the simulation. The simulation should produce two graphs, similar to Figure 6.2.
4. Approximate the new allelic frequencies for A1 and A2 at generation 200 (to two decimals) for each population. Then approximate the three genotype frequencies. Record these frequencies in Table 6.1.
5. Repeat this procedure 4 more times. Take observational notes below Table 6.1 for each population’s simulation (e.g. How did the allelic frequencies change over time? Did any alleles get “fixed” or “lost” and at approximately what generation?)
| Table 6.1 Bottleneck simulation data table |
| Pop. # |
Starting Population Size |
Starting Allelic Frequency |
Bottleneck generation range |
Bottleneck Population Size |
Approx. Allelic Freq. at generation 200 |
Calculated genotypic frequency at generation 200 |
| |
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A1 |
A2 (a) |
Start |
End |
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A1 |
A2 (a) |
A1A1 |
A1A2 |
A2A2 |
| Example |
1000 |
0.5 |
0.5 |
50 |
100 |
50 |
0.17 |
0.83 |
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OBSERVATIONAL NOTES:
Results & Discussion
1. During your four-population simulation of 250 generations, what happened to the allelic frequencies during and after the bottleneck event?
During:
After:
2. Compare the pattern of change you recorded for p (A1) and q (A2) at 200 generations. Is there a consistent trend or do the changes suggest chance events?
3. For any one of the populations in Table 6.1, convert your genotype frequencies to actual #s of individuals (by multiplying each by 1000), then input these data into Excel and run a chi squared test to compare the beginning and ending populations. What population did you use and what is your p-value? Are the populations significantly different?
4. Did the A1 allele become “fixed” or “lost” in any of the populations? What happens to the homologous allele when one allele becomes “fixed” or “lost”?
5. Alter the model parameter for bottleneck population size, first increase it to 100, 200, and 500, running a couple simulations each time you change the bottleneck population size. Then decrease it to 25 and 10, running a couple simulations of each. Describe the general response in allelic frequencies under those different scenarios, noting any trends you observe.
Increasing to 100, 200, and 500:
Decreasing to 25 and 10:
6. Reset you bottleneck population size to 50, then alter the model parameter for the “end” generation, decreasing it from 100 (50 generation bottleneck), to 75 (25 gen. bottleneck), 60 (10 gen.), and 55 (5 gen.). Run a simulation a couple times each setting and describe the general response in allelic frequencies under those different scenarios, noting any trends you observe.
7. What combination of bottleneck population size and number of bottleneck generations is most likely to produce the most drastic changes in allelic frequencies? What combination will produce the least drastic changes?
8. Would you expect the starting allelic frequencies to affect the likelihood of an allele going to fixation? If so, which allele (higher or lower frequency) will likely be fixed?
Try running a simulation with A1=0.8 and a BN Pop. = 10 (“start”=50, “end”=100). Did your results agree with your prediction? Can you explain why it might not?
9. Since only chance events are responsible for the change in gene frequencies, would you say that evolution has occurred? Explain.
Experiment B. Simulation of Migration: Gene Flow
Introduction
The migration of individuals between populations results in gene flow. In a natural population, gene flow can be the result of the immigration and emigration of individuals or gametes (for example, pollen movement). The rate and direction of migration and the starting allelic frequencies for the two populations can affect the rate of genetic change. The migration you will simulate is called source-sink migration, which is effectively a one-way migration from a large source population to another isolated sink population (e.g. continent to island). The starting allelic frequencies can differ for the two populations, and the rate of migration can be anywhere from 0 to 1. Another type of migration involves equal exchange of alleles between populations (two-way migration), which will not be simulated here.
Hypothesis
As your hypothesis, either propose a hypothesis that addresses migration specifically or state the Hardy‑Weinberg theorem.
Prediction
Either predict equilibrium values as a result of Hardy‑Weinberg or predict the type of change that you expect to observe as a result of migration (if/then).
Procedure
1. Begin by setting the online simulator with the following parameters (note the population size of 3000 and only 100 generations). You will need to click on the “Migration?” box twice to select source-sink migration. You will be modeling four populations at once, therefore you will only see allelic frequencies displayed (A1 in the top graph and A2 in the bottom graph), no genotype frequencies.
2. Run the simulation once with these original settings. Record what you observe happening to the allelic frequencies in the first row of Table 6.2.
3. Then, set the migration “rate” to 0.01 and the “Freq. A1” to 0.8. This means that 1% of your population (the island) is provided by immigrants from the source population with A1 frequency of 0.8.
4. Run the simulation again and record what you observe happening to the allelic frequencies in the appropriate row of Table 6.2. Note the number of generations that occur before the allelic frequencies of your island populations approximate the allelic frequency of the source population.
5. Run the model several more times, but change the rate of migration to 0.02, 0.03, 0.04, and 0.05 each time. Record what you observe happening to the allelic frequencies in the proper row of Table 6.2. Note the number of generations that occur before the A1 allelic frequencies of your island populations approximate the allelic frequency of the source population.
6. Reset the migration rate to 0.02, then change the source allelic frequency (“Freq. A1”) to 0.01, 0.25, 0.5, 0.75, and 1. Record your observations for each simulation in Table 6.2, specifically noting how each allele frequency changed.
| Table 6.2 Observations during altered migration rate and altered source population allelic frequency |
| Migration rate |
Source Population A1 Frequency |
Observations |
| 0.00 |
N/A |
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| 0.01 |
0.8 |
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| 0.02 |
0.8 |
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| 0.03 |
0.8 |
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| 0.04 |
0.8 |
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| 0.05 |
0.8 |
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| 0.02 |
0.01 |
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| 0.02 |
0.25 |
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| 0.02 |
0.50 |
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| 0.02 |
0.75 |
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| 0.02 |
1 |
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Results, Discussion, & Analysis
1. When simulating source-sink migration, what factors did you observe influencing the allelic frequencies of the sink population?
2. Imagine that you were using the Hardy-Weinberg Theorem as your null model while trying to detect the occurrence of source-sink migration between two populations. You would need to compare your observed frequencies to the expected frequencies using a chi-squared test. Explain the circumstances under which this method would not be able to detect migration between two populations, even a large amount of migration.
3. Run a simulation of only one population (not four), for only 10 generations , using a very high migration rate of 0.25 and a source population A1 frequency of 0.5 (keep the populations size of 3000 and A1 frequency of 0.5 for the sink populations). Record your model parameters and the actual (approximate) allelic frequency for A1 and A2 at the 1st generation in Table 6.3.
4. Record the expected allele frequencies and genotype frequencies for your population in Table 6.3, and then approximate the actual frequencies for each at the 2nd generation. Record in Table 6.3
5. Using Excel chi-squared, convert your your expected and actual genotypic frequencies into actual #s of individuals (multiply by your population size). Then run a Chi-squared test. Record the p-value below, then write a results sentence based upon this statistical test, and a conclusion statement about the existence of migration between those two populations based upon those results (NOT based upon your knowledge of whether migration was actually occurring):
P-value:
Results Statement:
Conclusion Statement:
6. Knowing the true migration rate, was the conclusion you made in Question 5 the correct conclusion? Why or why not?
7. Repeat steps 3-5 but set the model parameters for another simulation using a migrations rate of 0.25 and a source population A1 frequency of 0.8.
P-value:
Results Statement:
Conclusion Statement:
8. Was the conclusion you made in Question 7 the correct conclusion? Why or why not?
| Migr. rate |
Freq. A1 source pop. |
Starting Allelic Freq. in sink pop. |
Expected. Allelic Freq. |
Expected Genotypic Freq. |
Actual 1st Gener. Allelic Freq. |
Actual 1st Generation Genotypic Freq. |
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A1 |
A2 |
A1 (p) |
A2 (q) |
A1A1 (p2) |
A1A2 (2pq) |
A2A2 (q2) |
A1 (p) |
A2 (q) |
A1A1 (p2) |
A1A2 (2pq) |
A2A2 (q2) |
| 0.25 |
0.5 |
0.5 |
0.5 |
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| 0.25 |
0.8 |
0.5 |
0.5 |
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9. What has this simulation taught you about our ability to detect migration (a violation of an assumption for HW-Equilibrium) in natural populations?
Experiment C. Simulation of Natural Selection
Introduction
Natural selection, the differential survival and reproduction of individuals, was first proposed by Darwin as the mechanism for evolution. Although other factors have since been found to be involved in evolution, selection is still considered an important mechanism. 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 advantageous traits will increase in the next generation. The genotypic frequencies will change in the population. Whether traits are advantageous in a population depends on the environment and the selective agents (which can include physical and biological factors). Choose one of the following evolutionary scenarios to model natural selection in population genetics.
Figure 6.4. Two color forms of the peppered moth. The dark and light form of the moth are present in both photographs. Lichens are absent in (a) but present in (b). in which situation would the dark moth have a selective advantage?
Scenario 1. Industrial Melanism
The peppered moth, Biston betularia , is a speckled moth that rests on tree trunks during the day, where it avoids predation by blending with the bark of trees (an example of cryptic coloration). At the turn of the century, moth collectors in Great Britain collected primarily light forms of this moth (light with dark speckles) and only occasionally recorded rare dark forms (Figure 6.4). With the advent of the industrial Revolution and increased pollution, light‑colored lichens on the trees died, resulting in strong positive selection for dark moths resting on the now dark bark. The dark moth increased in frequency. However, in unpolluted regions, the light moth continued to occur in high frequencies. (This is an example of the relative nature of selective advantage, depending on the environment.) For this exercise, you will be simulating four populations of
Color is controlled by a single gene with two allelic forms, dark and light. Pigment production is dominant (A), and the lack of pigment is recessive (a). The light moth would be aa, but the dark form could be either AA or Aa.
Hypothesis
As your hypothesis, either propose a hypothesis that addresses natural selection occurring in polluted environments specifically or state the Hardy-Weinberg theorem.
Prediction
Either predict equilibrium values as a result of the Hardy‑Weinberg theorem or predict the type of change that you expect to observe as a result of natural selection in the polluted environment (if/then).
Procedure
1. To investigate the effect of natural selection on the frequency of light and dark moths, set the model to simulate 1 population, for 100 generations, and set the allelic frequencies of dark alleles A1 = 0.1. Click the box next to “Finite Pop.” to change the model to “Infinite pop.”, this will make a drop-down list show up. Be sure that your populations have A1 = 0.1 (population zero is the control, and should also be set to 0.1). With this setting, the frequency of the light allele A2 = 0.9 (not shown).
2. Assume that pollution has become a significant factor prior to this simulation, resulting in less light-colored lichen on the trees. In each population, a certain percentage of the light moths are eaten, but none of the dark moths are eaten. (In reality, some of each phenotype are eaten, but we will be using fitness values that are relative to the most fit phenotype.)
3. Set the fitness values of homozygous recessive genotypes (light colored moths) to A2A2 = 0.99. This means that only that proportion (e.g. 0.99) of the light moths will carry on from one generation to the next. Verify that your settings match these, then click the “OK” button:
4. Run the simulation and observe what happens. In Table 6.4, record your input parameters and approximate allelic and genotypic frequencies at the 80th generation.
5. Repeat this simulation, but change the fitness of A2A2 to 0.95, then 0.90, then 0.8. Record the information for each simulation in Table 6.4.
| Pop. #
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Starting Allelic Freq. in each pop. |
Fit-ness |
Genotype fitness |
80th Gener. Allelic Freq. |
80th Generation Genotypic Freq. |
| |
A1 |
A2 |
A2A2 |
A1A1 (dark) |
A1A2 (dark) |
A2A2 (light) |
A1 (p) |
A2 (q) |
A1A1 (p2) |
A1A2 (2pq) |
A2A2 (q2) |
| 1 |
0.1 |
0.9 |
.99 |
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| 2 |
0.1 |
0.9 |
.95 |
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| 3 |
0.1 |
0.9 |
.9 |
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| 4 |
0.1 |
0.9 |
.8 |
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6. While still working with an infinite population, set the model parameters to simulate for 200 generations, then set the fitness of A2A2 = 0.5 to simulate drastic selection against white moths. Run the simulation and see what happens.
a. Is the A2 allele (light, recessive) ever actually lost from this infinitely large population? Explain why this could theoretically occur.
7. Now, change the population to a finite population of 10,000 and run the simulation again.
a. Approximately how many generations did it take for the A2 allele to switch from 0.9 to 0.1?
b. Did the A2 allele become “lost”? If so, at approximately what generation?
8. Imagine that after many generations of pollution and lack of lichen on the trees, the allelic frequency has drastically shifted due to predation of moths. Now, the A1 allele (dark, dominant) has a frequency of 0.9 and the A2 allele (light, recessive) only has a frequency of 0.1.
a. Use the Hardy-Weinberg equation to calculate the genotypic frequencies expected with those allelic frequencies. Also calculate the number of moths of each color (phenotypes) that would occur in a population size of 10,000. Record these in Table 6.5 below.
b. Then, pollution is reduced and light-colored lichen returns to the trees. What phenotype and genotype of moth would be more conspicuous and likely to be selected against (lower fitness)?
c. Use the Hardy-Weinberg equation to calculate the genotypic frequencies expected with the “No Pollution” allelic frequencies in Table 6.5. Also calculate the number of moths of each color (phenotypes) that would occur in a population size of 10,000.
| Scenario
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Final Allelic Frequency |
Expected genotype frequency |
Expected phenotype prevalence |
| |
A1 |
A2 |
A1A1 (dark) |
A1A2 (dark) |
A2A2 (light) |
# Dark Moths |
# Light Moths |
| Pollution |
0.9 |
0.1 |
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| No Pollution |
0.1 |
0.9 |
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9. Re-run the model to select against the dark alleles with the frequency of A1 = 0.9, the fitness of A1A1 and A1A2 each set to 0.5, and with A2A2 fitness = 1.
a. Approximately how many generations did it take for the A1 allele to switch from 0.9 to 0.1? Was that faster of slower than when A2 was being selected against (question 7a)?
b. Did the A1 allele become “lost”? If so, at approximately what generation?
c. Think about these dominant (A1) and recessive (A2) alleles, and how the fitness values differed for each genotype in the scenario from step 9 and step 7. Explain why, when selected against, the A1 allele frequency declines and is “lost” faster than A2 was lost in your simulation from step 7.
d. Can you come up with a reason why the step 7 scenario is relevant when considering human health?
10. Would you say that evolution has occurred during the simulations in this exercise? Explain.
Bottle Neck simulations require you to set a starting generation, ending generation, and bottle neck population level
Two types of migration are possible: island or source-sink. When source-sink is selected, rate of migration and source pop. A1 frequency can be set.
Number of populations (1-5) to simulate at one time. Number of generations to simulate.
Check box to switch between “finite” and “infinite” population sizes, population size setting (finite populations), and starting allelic frequency for the A1 (dominant) allele.
Fitness of different genotypes. At least one genotype must be set to 1, the others are then fitness relative to the most fit genotype
More information in case you are lost
“Go” button begins the model, then changes to a “Reset” button, which can be pressed at any time to stop the model.
Graphs: When simulating only one population, the top graph will display the A1and A2 allelic frequencies, and the bottom will display each genotypic frequency.
When multiple populations are simulated, A1 frequency is shown on the top graph and A2 frequency is shown on the bottom
Information about if & when an allele is lost from a populations will be displayed to the right of the graph
Figure 6.2 Example output for A1 frequency from a bottleneck effect simulation of five populations .
Fig. 6.3 Source-sink migration from a continent to an island
Table 6.3 Expected and Actual allelic and genotypic frequencies
Table 6.4 Data from natural selection simulation
Table 6.5 Expected genotype frequency and phenotype occurrence when light-colored lichen is not prevalent.
Bio 112 Bignami & Olave Spring 2016
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