The Roll of a Million Dice

For our final discussion assignment, we'll be exploring the Hardy-Weinberg equation and its meaning in understanding the genetics of populations.

This equation was created as a simple logical interpretation of the ways the alleles of a gene should interact and combine in a completely randomly breeding population in which allelic frequencies are constant. There's no blinding biological insight behind it. If there are only two alleles available for a gene, whose frequencies are represented as p [f(A)] and q [f(a)], then the frequencies of those alleles added together (p + q) must equal 1 (100%).

And if these alleles are randomly being traded and recombined in the population, everyone having two alleles for the gene, then the resultant frequencies of the allelic combinations should be represented by the simple quadratic equation, (p + q) x (p + q) = 1, or p2 + 2pq + q2 = 1, where p2 = f(AA), 2pq = f(Aa) and q2 = f(aa). Because 1 x 1 = 1, these three expressions added together must also = 1, as all we did was square the (p + q) expression.

So now we want to take a look at some of the conditions required for a population to be in "Hardy-Weinberg equilibrium." Note, by the way, that such a population is actually impossible. We'll see why in a minute.

1.
Three of the conditions of the Hardy-Weinberg equilibrium are (1) mating must be random, (2) there must be no mutation, and (3) there must be no genetic drift. Clearly, the first of these conditions is obvious from the discussion of the H-W equation above. If the population isn't remixing genes randomly, you can't apply this kind of statistical evaluation. But after our discussions about mutation, it should be evident that the second condition is a problem. Why? Finally, explore the meaning of the last condition. What is genetic drift? What are founder effect and bottlenecking? Why does a large population size insulate a population against genetic drift? And why can genetic drift never truly be reduced to zero?

2.
Get out your calculator for the next two questions. Another restriction on Hardy-Weinberg populations is (4) that there be no migration going on. This is completely different from the genetic drift restriction you explored above. Migration is the situation in which a small population joins a larger population, combining its members (and their genes) with those of the original population. Why could this be a problem for Hardy-Weinberg equilibrium?

Consider a population of gerbils in which the frequency of black gerbils (B=Brown-dominant; b=black-recessive) is 1% [f(black gerbils)=0.01]. This population has been breeding randomly for several generations (in other words, you may assume that its genotypic frequencies represent a H-W equilibrium). A local stream temporarily dries up, allowing a second, smaller population of gerbils from the other side of the stream to join with our original population. In this population of gerbils, the frequency of black gerbils is 9% [f(black gerbils)=0.09]. The original gerbil population numbered 140 gerbils; the new members total 60 gerbils. The gerbils mix, get friendly, and begin to breed with each other. After a couple of generations of random mating, what is the new frequency of black gerbils in the combined population? Don't forget to include your work.

3.
The final restriction on Hardy-Weinberg populations is selection. (5) There must be no natural or artificial selection at work in the population. Again, this is a different issue from the random selection of mates. Selection can occur at all kinds of levels. Sometimes it's complete (ie, one phenotype is completely prevented from breeding), but more often it's partial. Both of these can be calculated, but complete selection is much simpler.

In a population of snap dragons, one flower color gene has two alleles, red and white. These two alleles show incomplete dominance--the heterozygote has pink flowers. In this population, twenty percent of the alleles in the population are white alleles [q=0.20]. First, calculate the f(pink flowering plants) for this original population. Then assume that a disease sweeps through this flower bed, killing all white-flowering plants before breeding season. After the remaining plants mate randomly, what will be the new f(pink flowering plants)? Don't forget to show your work. One final question. In both the migration and the selection problems above, I specified that at least one generation of random mating had occurred before I asked you to calculate new frequencies. Why did I make this stipulation?

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