Genetic Theory Pak Sham SGDP Io P London

Genetic Theory Pak Sham SGDP, Io. P, London, UK

Interpretation Theory Inference Model Formulation Data Experiment

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

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

Two offspring Sib 2 AA S i b 1 AA AA AA AA AA AA AA AA AA AA

IBD sharing for two sibs AA AA AA 2 1 1 0 AA 1 2 0 1 AA 1 0 2 1 AA 0 1 1 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

IBS IBD A 1 A 2 A 1 A 3 IBS = 1 IBD = 0 A 1 A 2 A 1 A 3

X 1 Y via X : 5 meioses via Y : 5 meioses 2 - identify all nearest common ancestors (NCA) - trace through each NCA and count # of meioses - expected IBD proportion = (½)5 + (½)5 = 0. 0625

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

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

Recombination & map distance Haldane map function

1 Segregation of three linked loci 2 (1 - 1)(1 - 2) (1 - 1) 2 1(1 - 2) 1 2

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

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

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

Population Frequencies Single locus Allele frequencies A P(A) = p a P(a) = q AA p(AA) = u Aa p(Aa) = v aa p(aa) = r Genotype frequencies

Mating type frequencies u v r AA Aa aa u AA u 2 uv ur v Aa uv v 2 vr r aa ur vr r 2 Random mating

Hardy-Weinberg Equilibrium u+½v A r+½v a u+½v r+½v A a 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 + ½ 2(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

Hardy-Weinberg frequencies Genotype frequencies: AA p(AA) = p 2 Aa p(Aa) = 2 pq aa p(aa) = q 2

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

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

Haplotype frequency table Locus B Locus A B b A pr+D ps-D p a qr-D qs+D q r s Dmax = Min(ps, qr), D’ = D / Dmax R 2 = D 2 / pqrs

Causes of allelic association Tight Linkage Founder effect: D (1 - )G Genetic Drift: R 2 (NE )-1 Population admixture Selection

Genotype-Phenotype Relationship Penetrance = Prob of disease given genotype AA Aa aa Dominant 1 1 0 Recessive 1 0 0 General f 2 f 1 f 0

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

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

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

Components of variance Phenotypic Variance Environmental Genetic Gx. E interaction

Components of variance Phenotypic Variance Environmental Additive Genetic Dominance Gx. E interaction Epistasis

Components of variance Phenotypic Variance Environmental Additive Genetic Dominance Quantitative trait loci Gx. E interaction Epistasis

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

QTL Variance Components Additive QTL variance VA = 2 p(1 -p) [ a - d(2 p-1) ]2 Dominance QTL variance VD = 4 p 2 (1 -p)2 d 2 Total QTL variance VQ = V A + V D

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

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

Average correlation in QTL effects MZ twins P( =0) P( =1/2) P( =1) =0 =0 =1 Average correlation Additive component = 0*0 + 0*1/2 + 1*1 =1 Dominance component = 0*0 + 1*1 =1

Average correlation in QTL effects Sib pairs P( =0) P( =1/2) P( =1) = 1/4 = 1/2 = 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

Decomposing variance E Covariance A C 0 Adoptive Siblings 0. 5 DZ 1 MZ

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)

Variance components Unique Environment Shared Environment E Additive Genetic Effects C e A c a Dominance Genetic Effects D d Phenotype P = e. E + a. A + c. C + d. D

ACE Model for twin data 1 [0. 5/1] E C e c PT 1 A a A C a c PT 2 E e

QTL linkage model for sib-pair data 1 [0 / 0. 5 / 1] N S n s PT 1 Q q Q S q s PT 2 N n

Population sib-pair trait distribution

Under linkage

No linkage

Interpretation Theory Inference Model Formulation Data Experiment