A flask of nutrient broth, buffered to maintain pH, is inoculated with a strain of E. coli. The flask is placed in a constant temperature environment where it is aerated by shaking.

1. Predict the effect of a change in energy availability over time.
2. Represent the change graphically in terms of the number of cells as a function of time.
3. In your graph as time progresses there is a change in the growth rate of the population. Add annotation to your graph to describe the time interval during which the growth rate is increasing linearly in proportion to the number of cells. Add annotation to your graph to describe another time interval during which the growth rate is decreasing in proportion to the square of the number of cells. Add a third annotation to describe an interval of time where the rate of growth is zero.
4. Select and justify two measurements of the E. coli population that could be made at two different points in time during growth that would be sufficient to answer questions about the population size at any time.
5. Describe the population of E. coli if the environment was continuously supplement by additional nutrient broth.

The following problem extends the Hardy-Weinberg model of population dynamics that was covered in Chapter 19. It applies mathematics that would be appropriate after a second course in Algebra. While the concept applied in this problem are within the scope of the Exam the mathematical representations are not and the item is provided to allow students who are able to take another look at the concepts.

The Hardy-Weinberg model of population dynamics is an algebraic representation of the relationships among genotype frequencies (F), the probability of the dominant allele A (p), and the probability of the recessive allele a (q). The Hardy-Weinberg model of population dynamics is based on several assumptions. One of these assumptions is “random mating.” If all genes in a population are equally able to reproduce, this means that all genes are equally fit and equally fertile. Consequently, the population does not evolve while in Hardy-Weinberg equilibrium.

However, populations do evolve and the Hardy-Weinberg model can be modified slightly to allow evolution to occur. Suppose that there is an initial population at generation zero and the probability of the dominant allele at that time is p0. Later, at population k, the probability is different. If the frequencies of the three different combinations of alleles are known, then the probabilities pk and qk can be calculated at generation k.

Because p and q are probabilities for a case where only two alleles exist, p+q=1. Additionally, (p+q)²=1, leading to the Hardy-Weinberg equation

The gene distribution never changes and p_k=p_k-1.

The equations of the Hardy-Weinberg model were modified (Haldane, 1924) to create a model in which evolution occurs.

Haldane divides by the factor W=F_k(AA)+F_k(Aa)+F_k(aa) so that the probabilities that are still calculated with Exercise 36.129 to continue to satisfy the condition for p and q to represent probabilities: (p+q)²=1.

1. Justify Haldane’s model in terms of what the factors w_AA, w_Aa, and w_aa mean.
2. Suppose that w_AA = w_Aa = 1, but that w_aa = 0.8. Predict what will happen to the population over time.
   
    
   Fitness is determined by the environment. Moree (The American Naturalist, 86, 1952) measured the relative fitness in Drosophila melanogaster of a recessive allele that imparts black eye color as population density increases. A varying number of flies with an equal number of males and females were placed in a pint jar and progeny counted. In each experiment the population was initially heterozygous.
   
    

   Number of females x Number of males	waa
   1 x 1	0
   10 x 10	0.06
   50 x 50	0.11
   150 x 150	0.46

3. Apply Haldane’s approach to calculate the probability p in the first generation after mating 150 female and 150 male flies that are heterozygous using w_AA = w_Aa = 1.
   
    
   Rendel (Evolution, 5, 1951) conducted an investigation of the dependence of fecundity (fertility) on light in ebony-eyed D. melanogaster. A summary of some of the data that he reported is shown in the table below:
   
    

   	Fraction females inseminated
   Phenotype of male	Light condition	Dark condition
   Ebony	0.215	0.607
   Wild type	0.494	0.466

4. Pose two scientific questions concerning the behavioral response indicated by the data that can be tested experimentally.