Genes in Populations: Hardy Weinberg Equilibrium. Biostatistics 666

Transcription

1 Genes in Poulations: Hardy Weinberg Equilibrium Biostatistics 666

2 Previous Lecture: Primer In Genetics How information is stored in DNA How DNA is inherited Tyes of DNA variation Common designs for genetic studies

3 Recommended Reading Lander and Schork (1994) Genetic Dissection of Comlex Traits Science 65: Paer was written >15 years ago, well before the human genome was sequenced Now, studies made easier by: Availability of reference genome sequence Much imroved genotying and sequencing technologies

4 Imortant Issues to Consider What are the secific challenges of genetic studies in humans? What are common strategies for imroving the ower of a genetic study? How can we combine different strategies to achieve cost-effective studies?

5 Always Check Data Quality! Study results are only as good as the data available and as this examle shows things can go wrong!

6 Today Proerties of alleles in a oulation Allele frequencies Genotyes frequencies Hardy-Weinberg equilibrium

7 Alleles Alternative forms of a articular sequence Each allele has a frequency, which is the roortion of chromosomes of that tye in the oulation

8 Allele Frequency Notation For two alleles Usually labeled and q = 1 For more than alleles Usually labeled A, B, C... subscrits A, B and C indicate allele name

9 Genotye The air of alleles carried by an individual If there are n alternative alleles there will be n(n+1)/ ossible genotyes Homozygous Genotye Genotye where the two alleles are in the same state Heterozygous Genotyes Genotye where the two alleles are in different states

10 Genotye Frequencies Since alleles occur in airs, these are a useful descritor of genetic data However, in any non-trivial study we might have a lot of frequencies to estimate... AA, AB, AC, BB, BC, CC

11 The simle art Genotye frequencies lead to allele frequencies For examle, for two alleles: A = AA + ½ AB B = BB + ½ AB However, the reverse is also ossible!

12 Hardy-Weinberg Equilibrium Random union of gametes Relationshi described in 1908 Hardy, British mathematician Weinberg, German hysician Shows n allele frequencies determine n(n+1)/ genotye frequencies Large oulations

13 Required Assumtions Diloid, sexual organism Non-overlaing generations Autosomal locus Large oulation Random mating Equal genotye frequencies among sexes Absence of natural selection

14 Random Mating: Mating Tye Frequencies Mating Frequency A 1 A 1 *A 1 A 1 11 ² A 1 A 1 *A 1 A 11 1 A 1 A 1 *A A 11 A 1 A *A 1 A 1 ² A 1 A *A A 1 A A *A A ² Total 1.0

15 Mendelian Segregation: Offsring Genotye Frequencies Offsring Mating Frequency A 1 A 1 A 1 A A A A 1 A 1 *A 1 A 1 11 ² 11 ² A 1 A 1 *A 1 A A 1 A 1 *A A A 1 A *A 1 A 1 ² ¼ 1 ² ½ 1 ² ¼ 1 ² A 1 A *A A A A *A A ² ²

16 And now ) )( ( ' ) ( ' ) ( '

17 Conclusion Genotye frequencies are function of allele frequencies Equilibrium reached in one generation Indeendent of initial genotye frequencies Random mating, etc. required Conform to binomial exansion ( 1 + ) =

18 Simle HWE Exercise If the defective alleles of the cystic fibrosis (CFTR) gene have a cumulative frequency of 1/50 what is: The roortion of carriers in the oulation? The roortion of affected children at birth?

19 A few more notes Extends to multile alleles Exand ( k ) Frequency of A/A homozygotes is A Frequency of A/B heterozygotes is A B Holds in almost all human oulations Little inbreeding (tyical F = ~0.005)

20 Something to think about Why would inbreeding matter?

21 Checking Hardy-Weinberg Equilibrium A common first ste in any genetic study is to verify that the data conforms to Hardy-Weinberg equilibrium Deviations can occur due to: Systematic errors in genotying, Unexected oulation structure, Presence of homologous regions in the genome, Association with trait in case-control studies. Which of these causes would you exect to increase the roortion of heterozygotes?

22 Testing Hardy Weinberg Equilibrium Consider a samle of N alleles n A alleles of tye A n B alleles of tye B n AA genotyes of tye AA n AB genotyes of tye AB n BB genotyes of tye BB

23 Simle Aroach Calculate allele frequencies and exected counts Construct chi-squared test statistic Convenient, but can be inaccurate, esecially when one allele is rare

24 A Better Aroach: Exact Test of Genotyic Proortions Iterate over all ossible outcomes and sum robabilities of outcomes with equal or lesser robability One sided tests are also ossible Aroach analogous to Fisher s exact test for contingency tables P HWE * n AB I P( N AB n AB N, n a ) P( N AB n * AB N, n a ) P( N AB n * AB N, n a )

25 Probability of n AB Heterozygotes To reconstruct formula, first calculate: Possible rearrangements for N alleles Possible rearrangements with n AB heterozygotes P( N AB n AB N, n A ) n AA n! n AB AB N!! n BB! na! nb! N! Calculation can be carried out efficiently in recursive fashion

26 Probability of n AB Heterozygotes Possible rearrangements for N alleles!/!! Possible rearrangements with n AB heterozygotes!/!!! Ratio of the two numbers is

27 Exact Test Observed

28 Comarison of Test Statistics

29 Comarison of Tye I Error Rates

30 Tye I Error Rate Is Periodic! Exact test in red, ² test in blue

31 Poor Genotye Calling is the most common real life cause of deviations from HWE Most modern studies exclude markers that fail HWE tests In genomewide association studies, thresholds of < 10-3 to 10-6 are common

32 Examle of Good & Bad Genotye Calling

33 More Genotye Calling Examles

34 Summary Hardy Weinberg Equilibrium holds in most human oulation samles Deviations from HWE can indicate oulation structure or natural selection, but most often are genotying artifacts Exact tests for Hardy Weinberg equilibrium rovide accurate results and are tyically recommended

35 Recommended Reading Wigginton et al. (005) A note on exact tests of Hardy-Weinberg equilibrium. Am J Hum Genet 76:887-93

36 Reading for Next Lecture Cardon and Bell (001) Association study designs for comlex diseases. Nature Reviews Genetics :91-99 Surveys imortant issues in analyzing oulation data. Defines linkage disequilibrium.