Population Genetics Lab 2

1. Population Genetics Lab 2 BINOMIAL PROBABILITY & HARDY-WEINBERG EQUILIBRIUM

2. Last Week : Sample Point Methods: Example: Use the Sample Point Method to find the probability of getting exactly two heads in three tosses of a balanced coin. 1. The sample space of this experiment is: 2. Assuming that the coin is fair, each of these 8 outcomes has a probability of 1/8. 3. The probability of getting two heads is the sum of the probabilities of outcomes 2, 3, and 4 (HHT, HTH, and THH), or 1/8 + 1/8 + 1/8 = 3/8 = 0.375.

3. Sample- point method : Example: Find the probability of getting exactly 10 heads in 30 tosses of a balanced coin. Total # of sample points = 230 = 1,073,741,824

4. Need a way of accounting for all the possibilities Example: In drawing 3 M&Ms from an unlimited M&M bowl that is always 60% red and 40% green, what is the P(2 green)? If one green M&M is just as good as another…

5. Binomial Probability Distribution Where, n = Total # of trials. y = Total # of successes. s = probability of getting success in a single trial. f = probability of getting failure in a single trial (f = 1-s).

6. Assumptions of Binomial Distribution # of trials are independent, finite, and conducted under the same conditions. There are only two types of outcome.(Ex. success and failure). Outcomes are mutually exclusive and independent. Probability of getting a success in a single trial remains constant throughout all the trials. Probability of getting a failure in a single trial remains constant throughout all the trials. # of success are finite and a non-negative integer (0,n)

7. Properties of Binomial Distribution Mean or expected # of successes in n trials, E(y) = ns Variance of y, V(y) = nsf Standard deviation of y, σ (y) = (nsf)1/2

8. Example: Find the probability of getting exactly 10 heads in 30 tosses of a balanced coin. Solution: We know, n = 30 y = 10 s = 0.5 f = 0.5

9. Example: Find the expected # of heads in 30 tosses of a balanced coin. Also calculate variance. Solution: E(Y) = ns = 30*0.5 = 15 V(Y) = nsf = 30*0.5*0.5 = 7.5

10. Problem 1 (10 minutes) A nuclear allozyme locus has two alleles, A1 and A2, with frequencies 0.967 and 0.133, respectively. If we sample 30 diploid individuals, what is the probability of: Not finding any copies of A2? Finding at least one copy of A2? GRADUATE STUDENTS ONLY: Finding fewer than 2 copies of A2?

11. Example: How many diploid individuals should be sampled to detect at least one copy of allele A2 from Problem 1 with probability of at least 0.95? Solutions: Thus, to detect at least one copy of allele A2 with probability of 0.95, one would need to sample at least 90 alleles (i.e., at least 45 diploid individuals).

12. Problem 2 (15 minutes) A plane carrying 280 medical doctors crashes in a remote stretch of the Andes, and a passenger with type O-negative blood is in desperate need of a transfusion. Given that the frequency of this blood type is 0.066 in the general population, what is the probability that a suitable donor will be found among the remaining passengers? A small team hikes to a nearby village hoping to find a suitable donor. How large would the village have to be so that there would be a 95% probability of finding a donor? GRADUATE STUDENTS ONLY: What are the assumptions of these calculations? Propose a more proper way of doing this analysis.

13. Estimation of allele frequency for Co-dominant locus Where, p = Frequency of allele A1 q = Frequency of Allele A2 N11 = # of individuals with genotype A1A1 N12 = # of individuals with genotype A1A2 N22 = # of individuals with genotype A2A2 N = total # of diploid individuals =N11+N12+N22

14. Estimation of Standard Error Where, p = Frequency of allele A1 q = Frequency of Allele A2 SEp = Standard error for frequency of allele A1 SEq = Standard error for frequency of allele A2 N = total # of diploid individuals =N11+N12+N22

15. Standard Deviation v. Standard Error We expect ~68% of the data to fall within 1 standard deviation of the mean.

16. Example: What are the allele frequencies of alleles A1 and A2, if the following genotypes have been observed in a sample of 50 diploid individuals? Solution: N11 = 17, N12 = 23, and N22 = 10

17. q = 1 – p = 0.43

18. Problem 3 (10 minutes) Estimate the allele frequencies (include their respective standard errors) for alleles A1, A2, and A3 if the following genotypes have been observed in a sample of 200 individuals:

19. Problem 4 (10 minutes) Go to the “Genetics Home Reference” website (http://ghr.nlm.nih.gov) and use the search feature to find a condition caused by a dominant allele in humans. On the main description page, find the frequency of the condition in human populations. Assuming HWE and Mendelian inheritance of the disease, what is the frequency of the recessive allele in this population? What is the standard error of this estimate? How many affected children would you expect in the next generation? What are the assumptions of these estimates?

20. Hypothesis Testing • Hypothesis: Tentative statement for a scientific problem, that can be tested by further investigations. • Null Hypothesis(H0): There is no significant difference in observed and expected values. • Alternate Hypothesis(H1): There is a significant difference in observed and expected values. • Example: • H0 = Fertilized and unfertilized crops have equal yields • H1 = Fertilized and unfertilized crops do not have equal yields

21. Remember: In final conclusion after the experiment ,we either – "Reject H0 in favor of H1" Or “Fail to reject H0”,

22. Type I error: Error due to rejection of a null hypothesis, when it is actually true (False positive). Level of significance(LOS) (α) : Maximum probability allowed for committing “type I error”. At 5 % LOS (α=0.05), we accept that if we were to repeat the experiment many times, we would falsely reject the null hypothesis 5% of the time.

23. P- value: • Probability of committing type I error • If P-value is smaller than a particular value of α, then result is significant at that level of significance

24. Testing departure from HWE In a randomly mating population, allele and genotype frequencies remain constant from generation to generation. Ho= There is no significant difference between observed and expected genotype frequencies (i.e. Population is in HWE) H1= There is a significant difference between observed and expected genotype frequencies (i.e. Population is not in HWE)

25. HWE Assumptions • Random mating • No selection • Equal numbers of offspring per parent • All progeny equally fit • No mutation • Single, very large population • No migration

26. χ2 - test Where,

27. Example: A population of Mountain Laurel at Cooper’s Rock State Forest has the following observed genotype counts: Is this population in Hardy-Weinberg equilibrium ?

29. The critical value (Table value) of χ2 at 1 df and at α=0.05 is approx. 3.84. Conclusion: Because the calculated value of χ2 (1160.49) is greater than the critical value (3.84), we reject the null hypothesis and accept the alternative (Not in HWE).

30. Problem 5 (10 minute) Based on the observed genotype counts in Problem 3, test whether the population that had been sampled is in HWE. Critical Chi square values are given in the table to the right. Think carefully about which one you should use (Hint: How many parameters are estimated from the data when the allele frequencies of 3 alleles are estimated?). What are some possible explanations for the observed results?