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Genetic Theory

Genetic Theory. Pak Sham SGDP, IoP, London, UK. Inference. Interpretation. Formulation. Experiment. Data. Theory. Model. Components of a genetic model. POPULATION PARAMETERS - alleles / haplotypes / genotypes / mating types TRANSMISSION PARAMETERS

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Genetic Theory

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

  2. Inference Interpretation Formulation Experiment Data Theory Model

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

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

  5. AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA AAAA AA AA Two offspring Sib 2 AA AAAA AA AA AA AA AA S i b 1

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

  7. IBS  IBD A1A2 A1A3 IBS = 1 IBD = 0 A1A2 A1A3

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

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

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

  11. Recombination & map distance Haldane map function

  12. (1-1)(1-2) (1-1)2 1(1-2) 12 Segregation of three linked loci 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. Conditional distribution of  at maker given  at QTL  at QTL 0 1/2 1  at M 0 1/2 1

  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 frequencies A P(A) = p • a P(a) = q • Genotype frequencies • AA p(AA) = u • Aa p(Aa) = v • aa p(aa) = r

  17. Mating type frequencies • uv r • AA Aa aa • u AA u2 uvur • v Aa uvv2 vr • r aa urvrr2 • Random mating

  18. Hardy-Weinberg Equilibrium • u+½v r+½v • A a • u+½v A • r+½v a u1 = (u0 + ½v0)2 v1 = 2(u0 + ½v0) (r0 + ½v0) r1 = (r0 + ½v0)2 u2 = (u1 + ½v1)2 = ((u0 + ½v0)2 + ½2(u0 + ½v0) (r0 + ½v0))2 = ((u0 + ½v0)(u0 + ½v0 + r0 + ½v0))2 = (u0 + ½v0)2 = u1

  19. Hardy-Weinberg frequencies • Genotype frequencies: • AA p(AA) = p2 • Aa p(Aa) = 2pq • aa p(aa) = q2

  20. Two-locus: haplotype frequencies • Locus B • B b • Locus A A AB Ab • a aB ab

  21. Haplotype frequency table • Locus B • B b • Locus A A pr ps p • a qr qs q • r s

  22. Haplotype frequency table • Locus B • B b • Locus A A pr+D ps-D p • a qr-D qs+D q • r s • Dmax = Min(ps,qr), D’ = D / Dmax • R2 = D2 / pqrs

  23. Causes of allelic association • Tight Linkage • Founder effect: D  (1-)G • Genetic Drift: R2 (NE)-1 • Population admixture • Selection

  24. Genotype-Phenotype Relationship • Penetrance = Prob of disease given genotype • AA Aa aa • Dominant 1 1 0 • Recessive 1 0 0 • General f2 f1 f0

  25. Biometrical model of QTL effects • Genotypic • means • AA m + a • Aa m + d • aa m - a 0 -a +a d

  26. Quantitative Traits • Mendel’s laws of inheritance apply to complex traits • influenced by many genes • Assume: 2 alleles per locus acting additively • Genotypes A1 A1 A1 A2 A2 A2 • Effect -1 0 1 • Multiple loci •  Normal distribution of continuous variation

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

  28. Components of variance • Phenotypic Variance • Environmental Genetic GxE interaction

  29. Components of variance • Phenotypic Variance • Environmental Genetic GxE interaction • Additive Dominance Epistasis

  30. Components of variance • Phenotypic Variance • Environmental Genetic GxE interaction • Additive Dominance Epistasis • Quantitative trait loci

  31. Biometrical model for QTL • Genotype AA Aa aa • Frequency (1-p)2 2p(1-p) p2 • Trait mean -a d a • Trait variance 2 2 2 • Overall mean a(2p-1)+2dp(1-p)

  32. QTL Variance Components • Additive QTL variance • VA = 2p(1-p) [ a - d(2p-1) ]2 • Dominance QTL variance • VD = 4p2 (1-p)2 d2 • Total QTL variance • VQ = VA + VD

  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  • 0 1/2 1 • Additive component 0 1/2 1 • Dominance component 0 0 1

  35. Average correlation in QTL effects • MZ twins P(=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 pairs P(=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 E Covariance A C 0 Adoptive Siblings 0.5 1 DZ MZ

  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 Dominance Genetic Effects Additive Genetic Effects Shared Environment Unique Environment E C A D e c a d Phenotype P = eE + aA + cC + dD

  40. ACE Model for twin data 1 [0.5/1] E C A A C E e c a a c e PT1 PT2

  41. QTL linkage model for sib-pair data 1 [0 / 0.5 / 1] N S Q Q S N n s q q s n PT1 PT2

  42. Population sib-pair trait distribution

  43. Under linkage

  44. No linkage

  45. Inference Interpretation Formulation Experiment Data Theory Model

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