INTRODUCTION

For the first three computer-lab assignments, you will be using a software package called "Populus" that was developed to provide simulations for mathematical problems in population genetics and ecology. The different programs operate similarly; all have certain parameters that you set yourself, before the computer generates the pattern of change over a set number of generations. The program provides you with these data in a series of graphs. For this and the upcoming assignments, you will be asked to enter certain parameter values and then interpret the graphs that the program will generate. You will also be asked to make up your own parameters and then explain the changes that you observe in the new graphs that are generated. Each assignment will be designed so that you can complete it in approximately one hour. 

Populus is free software, so you may download it and install it on your Windows PC. You may download it from the Deparatment of Biological Science server (recommended) at http://bio.fsu.edu/populus or from the Populus web site (http://biosci.cbs.umn.edu/software/populus.html). If you are going to do these assignments in the Biological Science computer lab, please keep in mind that many of your classmates might also want to do so. You should therefore start the assignments as soon as possible. Populus is installed on all PC's in the Conradi Rm 223 computer lab. You can find it on the desktop and under Start->Applications. Assignment #4 uses proprietary Macintosh software, so you will have to do this one in the Conradi room 217 computer lab.

The Populus programs are especially helpful because they contain a great deal of instructional text that will guide you through the exercise as well as be a helpful supplement to the lectures and the textbook material. Feel free to read as much of their text as you wish--even material that is not assigned to you. It might be fun, and it certainly won't hurt you.

In these assignments you should follow the directions provided here, which are devised to help you learn how population genetic dynamics occur. Feel free to explore and play with numbers to your heart's content; the more you play around, the better your intuitive grasp of population genetics will become. However, because this isn't all play, there are certain things we want you to do and which will serve as the basis for your grade. These will be numbered commands in all upper-case letters; the numbers will run sequentially within each subsection. When you turn in your assignment you should label your answers correspondingly. For example, a command in this section would be

1. WRITE YOUR NAME AND STUDENT NUMBER IN THE TOP LEFT-HAND CORNER OF YOUR ANSWER SHEET.

and you would turn in a sheet that had your name and number in the top left-hand corner, followed by "INTRODUCTION" (to let us know which section your answer addresses) and the number of the question/request. In some cases you might need to print a graph (to do so, when the graph is on the screen in front of you, hold down the ALT key while you press "p"). Any graph you turn in should be clearly labeled also.

HARDY-WEINBERG EQUILIBRIUM

Remember that, if a population is in Hardy-Weinberg equilibrium, the genotypic frequencies in the population can be predicted from the initial allele frequencies. You can run a quick simulation to show that this is true. Select Selection from the main menu by using the arrow keys to move up or down until it is highlighted then hit Enter. Now select Autosomal Selection and hit Enter. Hit Enter again to skip all the text here (you can read it next time when we simulate selection, in Assignment 2). Now you should be at the screen where you can set the parameters for the simulation. Use the arrow keys to move around on this screen. Highlight p vs t for the plot you want to view. Make sure that Fitness is highlighted on question 2. Set the number of generations to 100.  For the next set of parameters, w = fitness and we want no selection for this simulation (because Hardy-Weinberg assumes no selection, remember), so set the fitness, wAA, wAa, and waa each equal to 1. Next question, begin with a Single Frequency, which will allow you to set the initial frequency of p. To begin with, set it at 0.25. Now hit enter, and you're off and running. Use the space bar to see the various kinds of graphs. The first graph will show you the change in the frequency of p across generations. The second graph will show you the change in genotypic frequencies across generations. Are the frequencies for each genotype equal to those predicted by the Hardy-Weinberg equation? Don't worry about the last two graphs. When you've finished looking at the first two, hit Enter to get back to the parameter screen. Now try some different initial values of p to convince yourself that Hardy-Weinberg really works for an arbitrary initial value.

Now, what happens if we violate one of the assumptions of Hardy-Weinberg by imposing selection? To do so, change the fitness values of the three genotypes so that the AA homozygote is the favored genotype. Do so by setting wAA = 1.0, wAa = 0.0, and waa = 0.0. This is the mathematical equivalent to a situation in which all individuals that carry the Aa genotype or the aa genotype die before reaching maturity and thereby leave no offspring. Now run the program. What happens to the frequency of p across generations and the frequency of the three genotypes? Examine the graph that shows the change in frequencies of the three genotypes (by holding the Alt key while pressing "p").

1. DESCRIBE IN A FEW WORDS THE ULTIMATE FATE OF EACH GENOTYPE AS REVEALED BY THAT GRAPH.

GAMETIC DISEQUILIBRIUM

Hardy-Weinberg equilibrium can be established in one generation as long as we consider each single gene locus separately without being concerned about what is happening at other gene loci. This is not, however, always a realistic situation. This simulation will investigate the attainment of equilibrium when two gene loci, Aa and Bb, are segregating simultaneously.

For two loci, A and B, four types of gametes can be produced: AB, Ab, aB, and ab. We can divide these gametes into two types: "coupling" gametes (AB and ab) and "repulsion" gametes (Ab and aB). When the gamete frequencies are in equilibrium, the frequencies of the repulsion gametes are equal to the frequencies of the coupling gametes, and the products of the two types of gametes are equal at equilibrium: (Ab) x (aB) = (AB) x (ab). A difference between the products of the coupling and repulsion gametes indicates gametic disequilibrium, and the frequencies of the two-locus genotypes in the diploid zygotes will not be simple functions of the single-locus genotype frequencies. This difference in the products of coupling and repulsion gametes is a handy index for the level of disequilibrium.

Another way to think about these numbers is that difference between the products of the frequencies of coupling and repulsion gametes represents the change in gametic frequencies that must occur for equilibrium values to be reached. We will denote the difference between the products as D. D is a measure of the disequilibrium, and when D = 0, then the gametic frequencies are in equilibrium. If D is positive, so (AB)(ab) - (Ab)(aB) > 0, then at equilibrium, this fraction will have been subtracted from the frequencies of each of the coupling gametes and added to each of the repulsion gametes to make their products equal. If D is negative, the reverse will be true. The attainment of equilibrium between two loci (D = 0) will be slowed if linkage exists between the loci. The degree of linkage between two loci is given by r, the recombination rate. If r = 0.5, then the two loci are unlinked and assorting completely independently of one another. If r = 0, then the two loci are completely linked and undergo no independent assortment. The tighter the linkage, the longer it will take for the frequency of the coupling gametes to equal the frequency of the repulsion gametes. In other words, such linkage disequilibrium depends on recombination frequency, and lower recombination frequencies (r's closer to 0) between linked loci delay the attainment of equilibrium accordingly. This does not mean that the eventual equilibrium values for linked genes will be any different from those attained in the absence of linkage; D depends on gametic frequencies and not on linkage. Thus, once equilibrium is attained there is no way of distinguishing linked from unlinked genes except through tests for departures from independent assortment.

Remember that gametic disequilibrium can occur even between unlinked genes. Disequilibrium is a statistical phenomenon whose origins in biology we discussed in class (remember, when the number of individuals is less than the number of multilocus genotypes that is possible?). Linkage determines how rapidly disequilibrium decays, not how likely it is to be generated (we didn't prove this last point in class, but I can do so if you wish). 

Gametic Disequilibrium Where Loci Are Not Linked

Now let us examine some simulations. First, let's generate a simulation where the gametes are in equilibrium. Press escape (the Esc button on the upper left of the keyboard) to exit the Selection on a Single Autosomal Locus program. Now cursor down with arrow keys and highlight Selection on Two Loci; hit Enter. Set output plot on p vs t. Set the initial values of each gamete to 0.25. Set the recombination rate (r) = 0.5. We will assume no selection to begin with, so set w for each genotype = 1.0. Do 100 generations. Now run the simulation. The first graph shows the frequency of each gamete type across generations. Do the frequencies change? The second graph shows you the allele frequencies across generations. Do they change? The third graph shows you the value of D (the measure of disequilibrium) over generations. What is the value of D? Are the gamete frequencies in equilibrium? The last graph shows average fitness, which you set at 1.0.

Now, let's consider an extreme case of gametic disequilibrium, where you begin a population with the genotypes AABB and aabb. Thus the only gametes produced are AB and ab. Set these parameters PAB = 0.5, PAb = 0.0, PaB = 0.0, and Pab = 0.5. Then run the simulation. How many generations does it take for gamete frequencies to reach equilibrium? What is the initial value of D? How many generations did it take to drive D to 0?

Set gamete frequencies of your choice so that D is not zero and allow the simulation to run for 500 generations. Remember that D = (AB)(ab) - (Ab)(aB) and that the frequencies of A + a = 1.0 and B + b = 1.0. Examine the graph of gamete frequencies with time (counted in generations) and the graph of D as a function of time.

1. CHOOSE FIVE SETS OF INITIAL GAMETE FREQUENCIES, ALL OF WHICH CREATE AN INITIAL VALUE OF D THAT IS NOT ZERO. LIST EACH SET OF FREQUENCIES, AND STATE HOW MANY GENERATIONS ELAPSE BEFORE D = 0. COMMENT UPON WHETHER YOU FIND ANY PATTERN IN HOW MANY GENERATIONS IT TAKES FOR D TO REACH 0 AS A FUNCTION OF ITS INITIAL VALUE. NOTE: IF YOU CHOOSE A SET OF GAMETE FREQUENCIES THAT VIOLATE THE "EQUALITY RULES" (e.g. your frequencies do not sum as required) THE POPULUS PROGRAM WILL OVERRIDE YOUR CHOICE AND MAKE UP ITS OWN VALUES, BUT IT WON'T TELL YOU IT'S DONE THAT IN AN OBVIOUS MANNER. WHEN YOU TELL US YOUR VALUES, WE WILL THEREFORE MARK YOUR ANSWER "WRONG" EVEN THOUGH YOU MAY HAVE OBTAINED AN APPARENTLY SENSIBLE-LOOKING GRAPH. BE CAREFUL!

Linkage and Gametic Disequilibrium

Now let's introduce linkage between the two loci and see how it can generate disequilibrium and affect its rate of decay. Start with an example of disequilibrium by setting the following gamete frequencies: PAB = 0.05, PAb = 0.45, PaB = 0.45, and Pab = 0.05. Thus, (0.05)(0.05) is not equal to (0.45)(0.45), and D is negative, (0.05)(0.05) - (0.45)(0.45) = -0.2. Now introduce strong linkage, where r = 0.1 or 10 percent, and run teh simulation for 100 generations. How many more generations does it take to reach gametic equilibrium? to drive D to 0?

1. CHOOSE TWO SETS OF INITIAL GAMETE FREQUENCIES THAT PRODUCE A NON-ZERO D VALUE AND CHOOSE TWO VALUES OF THE RECOMBINATION RATE. EXAMINE THE TIME UNTIL THE INITIAL DISEQUILIBRIUM DISAPPEARED FOR ALL FOUR COMBINATIONS OF INITIAL GAMETE FREQUENCIES AND RECOMBINATION RATE. MAKE A 2-x-2 TABLE IN WHICH THE ROWS ARE THE INITIAL GAMETE FREQUENCIES THAT YOU CHOSE AND THE COLUMNS ARE THE TWO RECOMBINATION RATES; FOR EACH CELL WRITE THE INITIAL VALUE OF D AND THEN HOW MANY GENERATIONS IT TOOK FOR D TO REACH ZERO. EXPLAIN SUCCINCTLY THE ROLE OF EACH FACTOR IN DETERMINING THE RATE OF ATTAINMENT OF THE GAMETIC EQUILIBRIUM.

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 are finished with the computer please do a Start->Shutdown->Close all progams and log on as a different user.