Genetic Theory Pak Sham SGDP, IoP, London, UK

Theory Model Data Inference Experiment Formulation Interpretation

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 A A A A Paternal Maternal A A A A A A A A ½½ ½ ½ ¼ ¼ ¼ ¼

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

IBD sharing for two sibs AAAAAAAAAAAAAAAA AA AAAA AAAAAAAA AAAAAA 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 A1A2A1A2 A1A3A1A3 A1A2A1A2 A1A3A1A3 IBS = 1 IBD = 0

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 =

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

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

Recombination & map distance Haldane map function

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

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

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 frequenciesAP(A) = p aP(a) = q Genotype frequencies AAp(AA) = u Aap(Aa) = v aap(aa) = r

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

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

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

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

Haplotype frequency table Locus B Bb Locus AAprpsp aqrqsq rs

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

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

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

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

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

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

Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction

Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction Additive DominanceEpistasis

Components of variance Phenotypic Variance EnvironmentalGeneticGxEinteraction Additive DominanceEpistasis Quantitative trait loci

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)

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

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   01/21 Additive component01/21 Dominance component001

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

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

Decomposing variance 0 Adoptive Siblings 0.51 DZMZ A C E Covariance

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

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

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

Population sib-pair trait distribution

Under linkage

No linkage

Theory Model Data Inference Experiment Formulation Interpretation