1. Genetic Theory Pak Sham SGDP, IoP, London, UK

2. Theory Model Data Inference Experiment Formulation Interpretation

3. Components of a genetic model POPULATION PARAMETERS - alleles / haplotypes / genotypes / mating types TRANSMISSION PARAMETERS - parental genotype  offspring genotype PENETRANCE PARAMETERS - genotype  phenotype

4. Transmission : Mendel’s law of segregation A A A A Paternal Maternal A A A A A A A A ½½ ½ ½ ¼ ¼ ¼ ¼

5. Two offspring AAAAAAAAAAAAAAAA AA AAAA AAAAAAAA AAAAAA AA AA AAAA AA AA AAAA AA AA AAAA AAAA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA Sib 2 Sib1Sib1

6. IBD sharing for two sibs AAAAAAAAAAAAAAAA AA AAAA AAAAAAAA AAAAAA 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 Pr(IBD=0) = 4 / 16 = 0.25 Pr(IBD=1) = 8 / 16 = 0.50 Pr(IBD=2) = 4 / 16 = 0.25 Expected IBD sharing = (2*0.25) + (1*0.5) + (0*0.25) = 1

7. IBS  IBD A1A2A1A2 A1A3A1A3 A1A2A1A2 A1A3A1A3 IBS = 1 IBD = 0

8. 1 2 - identify all nearest common ancestors (NCA) X Y - trace through each NCA and count # of meioses via X : 5 meioses via Y : 5 meioses - expected IBD proportion = (½) 5 + (½) 5 = 0.0625

9. Sib pairs Expected IBD proportion = 2 (½) 2 = ½

10. Segregation of two linked loci Parental genotypes Likely (1-  ) Unlikely (  )  = recombination fraction

11. Recombination & map distance Haldane map function

12. Segregation of three linked loci (1-  1 )(1-  2 )  1  2 (1-  1 )  2  1 (1-  2 ) 1212

13. Two-locus IBD distribution: sib pairs Two loci, A and B, recombination faction  For each parent: Prob(IBD A = IBD B) =  2 + (1-  ) 2 =  either recombination for both sibs, or no reombination for both sibs

14. 0 1/2 1 0 1  at QTL  at M Conditional distribution of  at maker given  at QTL

15. Correlation between IBD of two loci For sib pairs Corr(  A,  B ) = (1-2  AB ) 2  attenuation of linkage information with increasing genetic distance from QTL

16. Population Frequencies Single locus Allele frequenciesAP(A) = p aP(a) = q Genotype frequencies AAp(AA) = u Aap(Aa) = v aap(aa) = r

17. Mating type frequencies u v r AAAaaa u AAu 2 uv ur v Aauv v 2 vr r aaur vr r 2 Random mating

18. Hardy-Weinberg Equilibrium u+½v r+½v Aa u+½v A r+½va u 1 = (u 0 + ½v 0 ) 2 v 1 = 2(u 0 + ½v 0 ) (r 0 + ½v 0 ) r 1 = (r 0 + ½v 0 ) 2 u 2 = (u 1 + ½v 1 ) 2 = ((u 0 + ½v 0 ) 2 + ½2(u 0 + ½v 0 ) (r 0 + ½v 0 )) 2 = ((u 0 + ½v 0 )(u 0 + ½v 0 + r 0 + ½v 0 )) 2 = (u 0 + ½v 0 ) 2 = u 1

19. Hardy-Weinberg frequencies Genotype frequencies: AAp(AA) = p 2 Aap(Aa) = 2pq aap(aa) = q 2

20. Two-locus: haplotype frequencies Locus B Bb Locus AAABAb aaBab

21. Haplotype frequency table Locus B Bb Locus AAprpsp aqrqsq rs

22. Haplotype frequency table Locus B Bb Locus AApr+Dps-Dp aqr-Dqs+Dq rs D max = Min(ps,qr), D’ = D / D max R 2 = D 2 / pqrs

23. Causes of allelic association Tight Linkage Founder effect: D  (1-  ) G Genetic Drift: R 2  (N E  ) -1 Population admixture Selection

24. Genotype-Phenotype Relationship Penetrance = Prob of disease given genotype AAAaaa Dominant110 Recessive10 0 Generalf 2 f 1 f 0

25. Biometrical model of QTL effects Genotypic means AAm + a Aam + d aam - a 0 d +a-a

26. Quantitative Traits Mendel’s laws of inheritance apply to complex traits influenced by many genes Assume: 2 alleles per locus acting additively GenotypesA 1 A 1 A 1 A 2 A 2 A 2 Effect -101 Multiple loci  Normal distribution of continuous variation

27. Quantitative Traits 1 Gene  3 Genotypes  3 Phenotypes 2 Genes  9 Genotypes  5 Phenotypes 3 Genes  27 Genotypes  7 Phenotypes 4 Genes  81 Genotypes  9 Phenotypes

28. Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction

29. Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction Additive DominanceEpistasis

30. Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction Additive DominanceEpistasis Quantitative trait loci

31. Biometrical model for QTL GenotypeAAAaaa Frequency(1-p) 2 2p(1-p)p 2 Trait mean-ada Trait variance  2  2  2 Overall meana(2p-1)+2dp(1-p)

32. QTL Variance Components Additive QTL variance V A = 2p(1-p) [ a - d(2p-1) ] 2 Dominance QTL variance V D = 4p 2 (1-p) 2 d 2 Total QTL variance V Q = V A + V D

33. Covariance between relatives Partition of variance  Partition of covariance Overall covariance = sum of covariances of all components Covariance of component between relatives = correlation of component  variance due to component

34. Correlation in QTL effects Since  is the proportion of shared alleles, correlation in QTL effects depends on   01/21 Additive component01/21 Dominance component001

35. Average correlation in QTL effects MZ twinsP(  =0) = 0 P(  =1/2) = 0 P(  =1) = 1 Average correlation Additive component = 0*0 + 0*1/2 + 1*1 = 1 Dominance component = 0*0 + 0*0 + 1*1 = 1

36. Average correlation in QTL effects Sib pairsP(  =0) = 1/4 P(  =1/2) = 1/2 P(  =1) = 1/4 Average correlation Additive component = (1/4)*0+(1/2)*1/2+(1/4)*1 = 1/2 Dominance component = (1/4)*0+(1/2)*0+(1/4)*1 = 1/4

37. Decomposing variance 0 Adoptive Siblings 0.51 DZMZ A C E Covariance

38. Path analysis allows us to diagrammatically represent linear models for the relationships between variables easy to derive expectations for the variances and covariances of variables in terms of the parameters of the proposed linear model permits translation into matrix formulation (Mx)

39. Variance components Phenotype A CE eac D d Unique Environment Additive Genetic Effects Shared Environment Dominance Genetic Effects P = eE + aA + cC + dD

40. ACE Model for twin data P T1 AC E P T2 ACE 1 [0.5/1] eaceca

41. QTL linkage model for sib-pair data P T1 QS N P T2 QSN 1 [0 / 0.5 / 1] nqsnsq

42. Population sib-pair trait distribution

43. Under linkage

44. No linkage

45. Theory Model Data Inference Experiment Formulation Interpretation