Part One: A Monte Carlo Simulation of Genetic Drift

Genetic drift is the random movement of allele frequencies between generations. More specifically, this movement is random with respect to the direction in which the allele frequencies change. Of course, the probabilities of movement in one direction or the other need not be equal, but even lopsided probabilities do not guarantee that in any particular population or in any particular generation the movement will be in one perfectly predictable direction. This randomness is in contrast to the effects of a deterministic force like natural selection in a very large population; when one allele is favored over another because of the increased fitness of phenotypes carrying that allele, the favored allele will always increase in frequency. This distinction is the reason we call genetic drift a stochastic phenomenon.

Two factors govern what happens in genetic drift. First, the effective population size determines the magnitude of this random change in allele frequencies in a single generation. Random sampling effects are more important in smaller populations because the probability that a sample of alleles will deviate in frequencies from the true frequencies is greater in small populations than in large ones. Second, the true allele frequencies determine the probability of movement in any particular direction. If one allele's true frequency is 0.90 and the other's is 0.10, a small sample is more likely to show an increase in the frequency of the common allele than a decrease.

Unless other forces are operating, drift leads inevitably to fixation of one allele. Let's investigate the influence of population size and initial allele frequencies on the cumulative changes in allele frequencies in the following simulations.

Begin Populus and select Genetic Drift from the first menu and "A Monte Carlo Model" from the second menu. Read the text for some basic information on how a Monte Carlo simulation works.

Now set the parameters for the model. Set population size at 100 and set the number of loci to simulate at 1. Set the initial frequencies at all six loci at "independently" and set the initial frequency of the "A" allele at locus 1 at a value of your choice. Tell the program you will not permit selfing and set the number of generations to run the simulation at 100 (use the right arrow key when you are highlighting the 3N value to move down to change the number of generations).

Run this simulation 10 times. For each run, record the number of generations until the "A" allele is either fixed (its relative frequency hits the value of 1.0) or goes extinct (its relative frequency drops to 0). If neither outcome has occurred after 100 generations, reset the number of generations to a higher number and rerun that simulation. I recommend that you start with 100 generations to make reading the graphs easier--if you start with a high number of generations you will have trouble reading precisely when fixation occurred.

1. WHAT WAS THE AVERAGE NUMBER OF GENERATIONS BEFORE THE "A" ALLELE BECAME EITHER FIXED OR LOST IN THE POPULATION IN THOSE 10 RUNS?

Now set the population size to 10 instead of 100 and run this simulation 10 times. Use the same initial frequency of the "A" allele at locus 1 that you used above when the population size was 100.

2. WHAT WAS THE AVERAGE NUMBER OF GENERATIONS BEFORE THE "A" ALLELE BECAME EITHER FIXED OR LOST IN THE POPULATION IN THESE 10 RUNS?
3. DESCRIBE HOW THE CHANGE IN THE POPULATION SIZE AFFECTED THE ANSWERS TO THE ABOVE QUESTIONS.

Now let's try looking at the pattern with 2 loci. Set population size = 100, number of loci to simulate = 2. Set the initial frequency of the "A" allele to 0.50 for each locus. Now run this simulation 10 times.

4. DO THE ALLELE FREQUENCIES OF THE "A" ALLELE CHANGE IN THE SAME DIRECTION AND AT THE SAME RATE FOR THE 2 LOCI IN EACH OF THE 10 RUNS? WHY OR WHY NOT?
5. IN HOW MANY OF THE 10 SIMULATIONS WAS "A" FIXED FOR LOCUS 1? FOR LOCUS 2? IN HOW MANY WAS THE "A" ALLELE FIXED SIMULTANEOUSLY FOR BOTH LOCI? WHAT WAS THE AVERAGE NUMBER OF GENERATIONS ELAPSED BEFORE FIXATION OF THE "A" ALLELE AT EACH LOCUS FOR THE 10 RUNS?

Now set the initial frequency of the "A" allele at 0.9 for locus 1 and 0.1 for locus 2. Run this simulation 10 times.

6. IN HOW MANY OF THE 10 SIMULATIONS WAS "A" FIXED FOR LOCUS 1? FOR LOCUS 2? HOW OFTEN WAS THE "A" ALLELE FIXED SIMULTANEOUSLY AT BOTH LOCI? WHAT WAS THE AVERAGE NUMBER OF GENERATIONS ELAPSED BEFORE FIXATION OF THE "A" ALLELE AT EACH LOCUS FOR THE 10 RUNS. HOW WAS THIS AVERAGE NUMBER DIFFERENT FROM THE AVERAGE CALCULATED IN QUESTION 2? WHY WAS IT DIFFERENT? HOW WILL LOW INITIAL ALLELE FREQUENCY AFFECT THE TIME TO FIXATION IN A SMALL POPULATION?

Part Two: Selection in Finite Populations

You will remember from the assignment on natural selection that, in an infinite population, the outcome of directional selection was always predictable; frequency of the favored allele increased until it became fixed in the population. In the previous section of this assignment, you learned that, in small populations, the change in allele frequencies across generations was unpredictable, and the fixation of one allele or the other occurred at random. Now we will investigate the interaction between natural selection and genetic drift in a finite population. Recall that selection is a predictable evolutionary force only when a population is sufficiently large or selection is sufficiently strong. The outcome of selection can be predicted by the relationship between the selection coefficient (s), a measure of the strength of selection against a genotype, and the effective population size (N).

• If s > 1/N, then natural selection will determine the outcome of allele frequencies
• If s < 1/N, then genetic drift will determine the outcome of allele frequencies

This is a powerful and important result. It means that, if an allele confers only a very small advantage in fitness over those of the other alleles at a locus, selection will be able to increase its frequency only if the population is very, very large. Thus not all favorable alleles will be spread through a population or species by natural selection, and good mutations of small effect may well be lost through genetic drift. Think about this problem from the other direction: in a small population only alleles with very large effects on fitness will be spread through natural selection. On the other hand, in large populations, even alleles with very small benefits will increase in frequency through natural selection. The purpose of this assignment is to convince you of this result and to allow you to explore these dynamics for yourself.

A. Weak Directional Selection in Finite Populations

Let's examine what happens in large and small populations when selection is weak. Select Genetic Drift from the first menu and Drift and Selection from the second menu. Read the text about selection and drift. Now set the parameters for the model. Set population size = 320. Set initial frequency of p = 0.1. Set genotype fitnesses: wAA = 1.0, wAa = 0.99, waa = 0.98. Set number of generations = 500. Now run this simulation 10 times. For each run, record the following data:

1. HOW MANY TIMES DOES "A" BECOME FIXED? HOW MANY TIMES DOES "A" GO EXTINCT?
2. DOES NATURAL SELECTION OR GENETIC DRIFT APPEAR TO BE THE DOMINANT FORCE IN THE EVOLUTION OF THIS POPULATION?

Now rerun this simulation 10 more times using the same parameters as in the first set of simulations but a smaller effective population size. Do 10 runs with N = 80 and 10 runs with N = 20. For each set of 10 runs record the following data:

B. Strong Directional Selection in Finite Populations

What happens if we increase the strength of selection favoring the "A" allele? Set initial frequency of p = 0.10. Set number of generations = 500. Set fitness values: wAA = 1.0, wAa = 0.90, waa = 0.80. Run 10 simulations at each of the following effective population sizes: N = 320, N = 80, and N = 20. For each of the 30 simulations record the following data:

1. HOW MANY TIMES DOES "A" BECOME FIXED? HOW MANY TIMES DOES "A" GO EXTINCT.
2. COMPARE THE NUMBER OF TIMES THAT "A" IS FIXED OR LOST AMONG THE THREE POPULATION SIZES.
3. IS THE STRONGER SELECTION FAVORING "A" ABLE TO OVERCOME GENETIC DRIFT WHEN N = 320? WHEN N = 80? WHEN N = 20?
4. GIVEN WHAT YOU HAVE LEARNED FROM THE PREVIOUS SECTION AND THIS ONE, DOES IT SEEM THAT ADAPTIVE EVOLUTION IS "UNSTOPPABLE" AS LONG AS FITNESS DIFFERENCES OCCUR? DOES EVOLUTION ALWAYS PRODUCE THE "BEST POSSIBLE SOLUTION" IN ALL SITUATIONS?

C. Natural Selection and Genetic Drift in Maintaining a Polymorphism

Remember that, when the heterozygote is the most fit genotype in a population, an equilibrium will be reached in allele frequencies and a stable polymorphism will be maintained. Let's investigate how population size influences the maintenance of such a polymorphism. We will start the population at the equilibrium allele frequencies by setting initial frequency of p = 0.5. Set relative fitnesses: wAA = 0.90, wAa = 1.0, waa = 0.90. Run for 500 generations. Do this simulation 10 times for each of the following effective population sizes: N = 400, N = 200, N = 100, N = 50. For each of the 40 runs record the following:

1. HOW MANY TIMES IS THE "A" ALLELE FIXED IN THE POPULATION? HOW MANY TIMES IS "A" LOST?
2. HOW DOES POPULATION SIZE INFLUENCE THE RATE AT WHICH "A" IS FIXED OR LOST (i.e. THE NUMBER OF GENERATIONS UNTIL FIXATION)?

Selection favoring the heterozygote in these populations should stabilize the frequency of the "A" allele at 0.50. FIND THE RANGE OF VARIATION IN THE RELATIVE FREQUENCY OF "A" THAT THE POPULATION "EXPLORES" AROUND 0.50 BY SUBTRACTING THE LOWEST RELATIVE FREQUENCY OF "A" FROM THE HIGHEST RELATIVE FREQUENCY OF "A" THAT OCCURS IN EACH SIMULATION.

3. CALCULATE THE AVERAGE RANGE IN THE RELATIVE FREQUENCY OF "A" FOR EACH OF THE EFFECTIVE POPULATION SIZES.
4. COMPARE AMONG THE FOUR EFFECTIVE POPULATION SIZES THE AVERAGE RANGE IN THE RELATIVE FREQUENCY OF "A" AND EXPLAIN HOW POPULATION SIZE INFLUENCES THIS RANGE.
5. IS THE POLYMORPHISM MAINTAINED MORE FREQUENTLY IN SMALL POPULATIONS OR LARGE POPULATIONS? DEFEND YOUR ANSWER WITH DATA FROM YOUR SIMULATIONS.
6. REVIEWING ALL THAT YOU SAW IN THIS EXERCISE, IF YOU WERE PRESENTED WITH A GRAPH OF THE FREQUENCY OF A PARTICULAR ALLELE FROM SOME NATURAL POPULATION MARCHING AROUND FROM ABOUT 0.15 TO 0.85, WOULD YOU BE WILLING TO DRAW ANY A PRIORI CONCLUSIONS ABOUT WHETHER THAT CHANGE WAS DRIVEN BY SELECTION OR DRIFT?   WOULD YOUR ANSWER CHANGE IF YOU KNEW THAT SUCH A CHANGE OCCURRED IN ONLY 15 GENERATIONS?  HOW ABOUT IF IT TOOK 150 GENERATIONS? IF YOU REFUSE TO DRAW A CONCLUSION, WHAT OTHER INFORMATION WOULD YOU WISH TO SEE BEFORE YOU WOULD BE WILLING TO VENTURE A CONCLUSION? DEFEND YOUR ANSWER.

Part Three: Some Final Thoughts

Let's consider briefly the likelihood of successful future adaptation in small and large populations. Remember that, in small populations, the likelihood of seeing a favorable mutation in any given generation is much smaller than that probability in large populations. So beneficial mutations will arise less often in small populations (assuming that mutation rates, per locus per gamete, are the same in the two types of populations). You have just seen that an allele must have a much larger effect on fitness in a small population in order for selection to increase its frequency. Now, if the distribution of allelic effects is such that mutations of small effect are more likely than mutations of large effect, then a greater fraction of beneficial mutations will be able to be incorporated into larger populations. On the whole, these results combine to suggest that the likelihood of successful future adaptation in small populations is very low indeed.

When you have finished with the assignment, hit escape until you return to the main menu. Cursor down to Exit Populus and hit Enter. If you have finished with the computer, please do a Start->Shutdown->Close all progams and log on as a different user.