Topic 2: Genetics and the logistic map

Attachment A4: Project option Topic 2: Genetics and the logistic map 2. Required background: 2.1 Science To complete this project you will require some background information on two topics: constrained population growth, and Mendelian genetics. Constrained population growth: In lectures we cover the geometric growth model, which is a simple discrete model of a changing quantity, commonly used to model populations. One of the weaknesses of the geometric model is that it assumes the population will continue growing indefinitely with the same growth rate. This is quite unrealistic over extended time periods, as factors such as competition for scarce resources will tend to reduce the growth rate. Thus there are two competing factors: as a population increases there are more individuals available for breeding, so there should be an increased growth rate, but there is also increased competition which will reduce the growth rate. The actual size of the population at any time will be determined by the interactions between these two factors (breeding and competition). This is an example of constrained growth. One of the most common models used for constrained population growth is the logistic model. In lectures near the end of semester we will study the logistic differential equation; in this project we will make use of the logistic map. Traditionally, logistic models are used to predict populations in closed ecosystems, such as fish in a lake. A key feature in logistic models is the carrying capacity of the ecosystem for a given species, which is the maximum number of organisms of that species which can be supported by the resources in the ecosystem (and is assumed to remain constant over time). In a logistic model, the population size will tend to approach the carrying capacity over time. Mendelian genetics: An elegant series of experiments was performed by Mendel, who revealed patterns governing the inheritance of physical characteristics in garden peas. He showed that some varieties always bred true to form. When he cross-pollinated plants with different characters, all the first-generation (F1) offspring only displayed one trait, the other apparently having disappeared. However, when he cross-pollinated the F1 hybrids, around one quarter of the second generation (F2) offspring displayed the missing trait. Mendel s observations led to the establishment of three rules of classical genetics: Physical traits are passed from parents to offspring by units of inheritance (genes); Offspring inherit two copies (alleles) of every gene, one from each parent; and Some genes are dominant, so are expressed even when the two alleles differ, and some are recessive, so are only expressed when the two alleles are the same. If an individual has two identical alleles of a gene then it is called homozygous for that gene, and if the alleles differ then it is called heterozygous. When considering population genetics, it should be apparent that genes and allele frequencies across the population can change over time, resulting in evolutionary change. However, the Hardy-Weinberg theorem states that the frequencies of alleles and genotypes in a population s gene pool remain constant from generation to generation whenever the following five conditions are met (which is uncommon in nature, except over short time periods): 1. Random mating (which is disrupted when individuals select mates, especially related ones); 1

2. Large population size (as small populations show chance fluctuations, known as genetic drift); 3. No allele transfer between populations; 4. No mutations (gene insertions, deletions, substitutions); and 5. No natural selection (that is, no differential survival or reproductive success of individuals). Such populations are then said to be in a state of Hardy-Weinberg equilibrium. Knowing that gametes (sperm and egg cells) carry only one copy of each chromosome, and therefore only one allele for each gene, allows predictions to be made about mating outcomes and the patterns of inheritance. By coding dominant alleles with capital letters (for example, P) and recessive alleles with lower-case letters (for example, p), it is possible to construct a 2 2 matrix, called a Punnett square to determine all possible combinations of alleles in offspring, and the relative frequencies of these combinations. For example, consider the inheritance of a single gene in a population of organisms, with the frequency of allele P equal to x = 0.8 and that of allele p equal to y = 0.2. Then we can calculate the relative frequencies of all three possible genotypes: the probability of homozygous PP is x 2 = 0.8 0.8 = 0.64. the probability of homozygous pp is y 2 = 0.2 0.2 = 0.04. the probability of heterozygous Pp or pp is 2xy = 2(0.8 0.2) = 0.32. This calculation can be completed in a Punnett square, as follows: Male parent P (x = 0.8) p (y = 0.2) Female P (x = 0.8) PP (x 2 = 0.64) Pp (xy = 0.16) parent p (y = 0.2) pp (yx = 0.16) pp (y 2 = 0.04) (Note that heterozygous Pp is identical to pp; each occurs with a relative frequency of 0.16, so the combined relative frequency is 0.32. Note also that the sum of the probabilities of each possible genoptype equals 1.) 2.2 Mathematics Let P be a population with a natural (unconstrained) growth rate of r per time period, let K be the carrying capacity and P i denote the population at time step i = 0, 1, 2,... (so P 0 is the initial population). The equation for the logistic map is ( ) K Pi P i+1 = P i + rp i. K ( ) K Pi The term rp i represents breeding, and the term represents competition. K 2.3 Python No additional Python knowledge is required. 3. Questions: Complete the following 14 questions. 3.1 Hand calculations Complete Questions 1 to 8 by hand, showing all working. Each question is independent of the others (so the carrying capacities and growth rates may change). For simplicity, in all of these calculations and your Python programs, do not round your numbers. Keep all numbers, including the number of fish, accurate to at least 4 decimal places. 2

1. The carrying capacity of a lake for a special species of Zebrafish is 10000 individuals, the unconstrained growth rate is 0.05 per month, and the initial population at time 0 is 1000 fish. (a) (1 mark) Estimate the population size at times i = 1 and 2 months; that is, find P 1 and P 2. (b) (1 mark) Repeat your calculation from Part (a), assuming now that 50 fish are harvested for pet shops just before the end of each month. (That is, the fish breed, then 50 are harvested, then the population is measured.) You may assume that the population of 1000 fish at time 0 is after the 50 fish have been harvested that month. 2. (1 mark) A lake with a carrying capacity of 10000 fish is mistakenly overstocked with 15000 fish. If the unconstrained growth rate is 0.1 per month, estimate the population size at times i = 1 and 2. 3. (4 marks) Biologists conduct three population surveys of a fish population at monthly intervals, showing that the population sizes are 5000, 5250 and 5500. Assuming the population follows a logistic curve, calculate the carrying capacity and unconstrained growth rate for this species. (Hint: use at least 5 significant figures in your calculations.) 4. (3 marks) The conditions for Hardy-Weinberg equilibrium assume random mating between geneotypes across the population. Whether or not this is a reasonable assumption in humans arguably depends on which characteristic is being considered. Briefly discuss (80 100 words) the assumption of random mating in humans. Your discussion must include two characteristics for which random mating is a reasonable assumption, and two for which it is less reasonable. 5. (1 mark) Assume that the colour of individual Zebrafish is determined by a single recessive colour gene, with homozygous cc individuals gold coloured, but other genotypes dark coloured. In a certain sample of fish, 60% of the individuals are homozygous CC and 10% of individuals are gold coloured. Calculate the relative frequency of allele c in this population. 6. The relative frequency of the allele c in a certain Zebrafish population that satisfies the conditions for Hardy-Weinberg equilibrium is 0.3. (a) (1 mark) Draw a Punnett square for the colour gene, and calculate the relative frequency of each genotype. What is the expected proportion of gold coloured fish? (b) (3 marks) Assume that gold coloured fish are popular with collectors but dark fish are not. If a number of gold fish are removed from the population each month, explain (in words) what will happen to the relative frequencies of the alleles for the colour gene over time? What will happen to the proportion of gold fish in the population after a large number of generations? Explain your answer briefly. Does the population remain in Hardy-Weinberg equilibrium? If not, which of the five conditions for Hardy-Weinberg equilibrium is/are no longer met? (c) (2 marks) If all of the gold fish are removed from a population that was previously in Hardy- Weinberg equilibrium, find the relative frequency of each allele of the colour gene in this generation and each genotype in the next generation. What proportion of individuals in the next generation will be gold? 7. Assume that Questions 1(a) and 6 refer to the same Zebrafish population. (a) (1 mark) Use your results from Questions 1(a) and 6 to estimate the number of gold fish in the lake at each of times 0, 1 and 2. (At this stage, do not assume that any fish are removed from the lake.) (b) (3 marks) A pet shop catches 100 fish at the end of each month, removes any gold fish they catch from the population and sells them, and returns any dark fish to the population. Assuming that the fish caught are completely random (so have relative frequencies of their genotypes equal to the those of the whole population), estimate the total fish population at time 1, directly after the pet 3

shop conducts its harvest for that month. (Note: the number of fish caught in this and subsequent questions is not the same as the number of fish sold. The number of fish sold is the expected number of gold fish amongst those that are caught.) (c) (1 mark) How many gold fish does the pet shop sell in this month? (d) (2 marks) What is the relative frequency of each allele of the colour gene and each genotype at time 1 (after the harvest), and how many fish in the lake will be gold? (e) (3 marks) Assuming the pet shop also catches 100 fish in the second month, how many of these are gold? At time 2 (after the harvest), what is the total fish population, the relative frequencies of alleles and genotypes in the population, and the number of gold fish remaining in the lake? 8. The pet shop has conducted a financial analysis of their operations, and they know that: It costs $1 for each fish caught per month. If the shop sells x gold fish in a month, then the price they receive per fish is $(150 10x), with a minimum selling price of $50. (So if they sell 3 fish the price per fish is $120, if they sell 5 fish the price is $100 per fish, and if they sell 12 fish then the price is $50 per fish.) For these calculations, ignore other financial factors such as tax and inflation. (a) (2 marks) Using this information and your result from Question 7(c), what profit does the shop make in the first month? (b) (2 marks) Using this information and your result from Question 7(e), what profit does the shop make in the second month? 3.2 Python programming 9. (a) (5 by 2 marks = 10 marks) Write each of the following Python functions in a file. A function called getallelefreq that: has three input parameters: the number of homozygous CC individuals in a population, the number of homozygous cc individuals and the number of heterozygous individuals; and returns the relative frequency of allele c in that population. A function called gethomcc that: has two input parameters: the relative frequency of allele c in a population and the population size; and returns the number of homozygous CC individuals in the population. A function called gethomcc that: has two input parameters: the relative frequency of allele c in a population and the population size; and returns the number of homozygous cc individuals in the population. A function called gethet that: has two input parameters: the relative frequency of allele c in a population and the population size; and returns the number of heterozygous individuals in the population. A function called MonthProfit that: has two input parameters: the number of gold fish sold that month and the total number of fish caught that month; and returns the profit for that month. 4

You do not need to print or submit these functions separately: they are used in Part (b), so you will submit them there. (b) (10 marks) Write a Python program which uses a logistic map to model populations of the coloured fish over time (so your program will need to include information about relative frequencies of alleles and genotypes). Your program must: Prompt the user to enter the following: the carrying capacity of the lake; the unconstrained growth rate of the fish per month; the initial fish population; the total number of months over which the population should be modeled; the initial relative frequency of the allele c in the population; the number of fish caught by the pet company each month (which might be zero); and whether the program should show a graph of the total fish population or of the gold fish population. Apply the logistic model to the population over the specified number of months. Each month allow the pet shop to catch the specified number of fish at random at the end of the month, with any caught gold fish removed from the population and all dark fish returned. Plot a graph of the fish population over the specified time period, with an appropriate title and labels on the axes. The user has a choice of plotting the total fish population or the gold fish population; At the end of the specified number of months, print the total number of fish in the final population, the total number of gold fish harvested, the number of gold fish in the final population, the proportion of gold fish in the final population and the total profit made by the pet shop. Have variables and functions with meaningful names, and be appropriately commented. Use the five functions from Part (a). (This is compulsory.) Hint(s): You may assume that all input values are valid (for example, the carrying capacity will be positive), and that the initial population and initial relative frequency of the allele is after any harvesting at time 0. (c) (4 by 2 marks = 8 marks) Test your program in the following ways. (You must include a printed copy of the output from your program in your submission.) (i) Use your program to repeat Question 1(a) from Section 3.1, and verify that the output matches your hand calculation. (ii) Use your program to repeat Question 2 from Section 3.1. (iii) Use your program to verify the values calculated in Question 3 from Section 3.1. (iv) Use your program to repeat Questions 7(b), 7(c), 7(d) and 7(e) from Section 3.1. (d) (6 marks) Write a brief user guide which explains how to use the program. (Your user guide should not assume that the user has read this assignment question sheet.) The guide should contain all necessary information about: What the program does. What input is requested by the program, and what valid input it can take. What output the program gives. Any assumptions you have made, or any special cases. This is a user guide, not a programmer s manual. Do not describe the algorithm you have used or internal details of the program. Instead, if someone with a basic understanding of computers (and 5

a good understanding of the science relevant to your project topic) wanted to run your program, what would they need to know? 3.3 Running the program (You must submit printed copies of your program and all graphs.) 10. Use your program to model a Zebrafish population with K = 10000 fish, r = 0.05 per month, P 0 = 1000 fish, relative frequency of allele c equal to 0.3 and zero fish caught per month, for a period of 100 months. (a) (1 mark) Plot a graph of the total fish population over time. (b) (1 mark) Plot a graph of the gold fish population over time. (c) (3 marks) What is the final number of fish, and the final proportion of gold fish in the population? Deduce what will be the proportion of each genotype in the final population, and explain your answer in terms of Hardy-Weinberg equilibria. What profit does the pet shop make? 11. Use your program to model a Zebrafish population with K = 10000 fish, r = 0.05 per month, P 0 = 1000 fish, relative frequency of allele c equal to 0.3 and 100 fish caught per month, for a period of 100 months. (a) (1 mark) Plot a graph of the total fish population over time. (b) (1 mark) Plot a graph of the gold fish population over time. (c) (2 marks) What is the final number of fish, and the final proportion of gold fish in the population? What profit does the pet shop make? (d) (2 marks) Compare your answers to Parts (a), (b) and (c) to the answers you obtained in Question 10. Briefly explain any differences. 12. Use your program to model a Zebrafish population with K = 10000 fish, r = 0.05 per month, P 0 = 1000 fish, relative frequency of allele c equal to 0.3 and 100 fish caught per month, for a period of 200 months. (a) (1 mark) Plot a graph of the gold fish population over time. (b) (3 marks) Briefly explain the shape of this graph, in particular why the graph initially rises and then drops off. Make reference to your answer to Question 6(b). 3.4 Written response 13. (20 marks) Select one of the following topics and write a 300 350 word response to the topic. Your response must identify which topic you chose, and must be written as an essay in an appropriate scientific style. If your topic asks you to state your opinion then make sure that you do so, with cogent arguments for and against your case. If relevant you can include some diagrams, equations or mathematics (which will not be included in the word limit), but your response should be predominantly text based. Your submission must be typed (although diagrams and calculations may be hand-written). The SCIE1000 Blackboard site contains a Criteria Sheet for this essay; you must print a copy of the sheet and attach it as the last page of your project submission. The sheet shows how marks will be allocated, to a maximum of 20 marks. For many years, commercially harvesting whales was a common and well-accepted practice in many countries. For example, Australia conducted commercial whaling until 1978; at Tangalooma on Moreton island (only 50 km from the University of Queensland), a whaling station operated from 1952 until 1962, and harvested around 6300 humpback whales. 6

In 1986, the International Whaling Commission announced a moratorium on commercial harvesting of whales. Many countries are signatories to that moratorium. However, there is ongoing international controversy about whaling, as some countries decline to be bound by the moratorium, or conduct scientific harvesting programs, or allow their indigenous population to harvest whales. This raises an interesting question. An argument that is commonly raised to support continued whaling and hunting of other species (including endangered species) is the cultural and historical significance of such hunting. (For example, indigenous Australians are allowed to hunt and kill sea turtles, even though the turtles are endangered.) How should human cultural beliefs and behaviours be balanced against environmental concerns? Is it always justified for a species to be hunted for special cultural reasons, or should there be limitations? A few years ago, koala populations were reaching very high numbers in a few locations of Australia, and were damaging their environment and food sources. There was a serious proposal to cull some of the koalas. At the same time, koalas are endangered or extinct in many other areas. In 2008, the Australian Department of Defence shot a large number of kangaroos at some of their sites, again after the population had increased very rapidly. Of course, both koalas and kangaroos are protected species. Culling animals (particularly cute species) is a very controversial activity. There was international outrage when the koala cull was proposed and when the kangaroo cull was undertaken. However, it was argued that there were strong scientific (and even moral) reasons for doing so. Discuss the issues associated with this. Is it appropriate to cull, should nature be allowed to take its course, or should some other action be taken? As a scientist, what advice would you provide? The end 7