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.

A. **Predict** the effect of a change in energy availability over time.

B. **Represent** the change graphically in terms of the number of cells as a function of time.

C. 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.

D. **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.

E. **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 another look at the concepts.

The Hardy-Weinberg model of population dynamics is an algebraic representation of the relationships among genotype frequencies, F, and the probability of the dominant allele A, p, and 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 never evolves.

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 p_{0}. Later, at population k the probability is different. But if the frequencies of the three different combinations of alleles is known then the probabilities p_{k} and q_{k} can be calculated at generation k

(1) $${p}_{k}={F}_{k}(AA)+\mathrm{\xbd}{F}_{k}(Aa)\phantom{\rule{0.25em}{0ex}}{q}_{k}={F}_{k}(aa)+\mathrm{\xbd}{F}_{k}(Aa)$$

And since p and q are probabilities for a case where only two alleles exist, p+q=1. Then also (p+q)^{2}=1, leading 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:

(2) $${F}_{k}(AA)={p}^{2}{}_{k}{w}_{AA}/W\phantom{\rule{0.25em}{0ex}}{F}_{k}(Aa)=2{p}_{k}{q}_{k}{w}_{Aa}/W\phantom{\rule{0.25em}{0ex}}{F}_{k}=$$

$${q}^{2}{}_{k}{w}_{aa}/W\phantom{\rule{0.25em}{0ex}}W={p}^{2}{w}_{AA}+2pq{w}_{Aa}/{q}^{2}{w}_{aa}$$

Haldane divides by the factor W=F_{k}(AA)+F_{k}(Aa)+F_{k}(aa) so that the probabilities that are still calculated with equation (1) to continue to satisfy the condition for p and q to represent probabilities: (p+q)^{2}=1.

A. **Justify** Haldane’s model in terms of what the factors w_{AA}, w_{Aa}, and w_{aa} mean.

B. 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 | w_{aa} |
---|---|

1 x 1 | 0 |

10 x 10 | 0.06 |

50 x 50 | 0.11 |

150 x 150 | 0.46 |

C. 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 |

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

E. Is there a question you can add here to wrap up this set with this LO from the list? In this case “light” is the single environmental factor, and they two phenotypes are ebony and wild type that result from different genotypes within the population of flies.