Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001

Linkage Mapping  Compares inheritance pattern of trait with the inheritance pattern of chromosomal regions  First gene-mapping in 1913 (Sturtevant)  Uses naturally occurring DNA variation (polymorphisms) as genetic markers  >400 Mendelian (single gene) disorders mapped  Current challenge is to map QTLs

Linkage = Co-segregation A2A4A2A4 A3A4A3A4 A1A3A1A3 A1A2A1A2 A2A3A2A3 A1A2A1A2 A1A4A1A4 A3A4A3A4 A3A2A3A2 Marker allele A 1 cosegregates with dominant disease

Recombination A1A1 A2A2 Q1Q1 Q2Q2 A1A1 A2A2 Q1Q1 Q2Q2 A1A1 A2A2 Q1Q1 Q2Q2 Likely gametes (Non-recombinants) Unlikely gametes (Recombinants) Parental genotypes

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

Map distance Map distance between two loci (Morgans) = Expected number of crossovers per meiosis Note: Map distances are additive

Recombination & map distance Haldane map function

Methods of Linkage Analysis  Model-based lod scores  Assumes explicit trait model  Model-free allele sharing methods  Affected sib pairs  Affected pedigree members  Quantitative trait loci  Variance-components models

Double Backcross : Fully Informative Gametes AaBb aabb AABB aabb AaBbaabb Aabb aaBb Non-recombinantRecombinant

Linkage Analysis : Fully Informative Gametes Count DataRecombinant Gametes: R Non-recombinant Gametes: N ParameterRecombination Fraction:  LikelihoodL(  ) =  R (1-  ) N Parameter Chi-square

Phase Unknown Meioses AaBb aabb AaBbaabb Aabb aaBb Non-recombinantRecombinant Non-recombinant Either : Or :

Linkage Analysis : Phase-unknown Meioses Count DataRecombinant Gametes: X Non-recombinant Gametes: Y orRecombinant Gametes: Y Non-recombinant Gametes: X LikelihoodL(  ) =  X (1-  ) Y +  Y (1-  ) X An example of incomplete data : Mixture distribution likelihood function

Parental genotypes unknown Likelihood will be a function of allele frequencies (population parameters)  (transmission parameter) AaBbaabb Aabb aaBb

Trait phenotypes Penetrance parameters Genotype Phenotype f2f2 AA aa Aa Disease Normal f1f1 f0f0 1- f 2 1- f 1 1- f 0 Each phenotype is compatible with multiple genotypes.

General Pedigree Likelihood Likelihood is a sum of products (mixture distribution likelihood) number of terms = (m 1, m 2 …..m k ) 2n where m j is number of alleles at locus j

Elston-Stewart algorithm Reduces computations by Peeling: Step 1 Condition likelihoods of family 1 on genotype of X. 1 2 X Step 2 Joint likelihood of families 2 and 1

Lod Score: Morton (1955) Lod > 3  conclude linkage Prior odds linkage ratioPosterior odds 1: :1 Lod <-2  exclude linkage

Linkage Analysis Admixture Test Model Probabilty of linkage in family =  Likelihood L( ,  ) =  L(  ) + (1-  ) L(  =1/2)

Allele sharing (non-parametric) methods Penrose (1935): Sib Pair linkage For rare diseaseIBD Concordant affected Concordant normal Discordant Therefore Affected sib pair design Test H 0 : Proportion of alleles IBD =1/2

Affected sib pairs: incomplete marker information Parameters: IBD sharing probabilities Z=(z 0, z 1, z 2 ) Marker Genotype Data M: Finite Mixture Likelihood SPLINK, ASPEX

Joint distribution of Pedigree IBD  IBD of relative pairs are independent e.g If IBD(1,2) = 2 and IBD (1,3) = 2 then IBD(2,3) = 2  Inheritance vector gives joint IBD distribution Each element indicates whether paternally inherited allele is transmitted (1) ormaternally inherited allele is transmitted (0)  Vector of 2N elements (N = # of non-founders)

Pedigree allele-sharing methods Problem APM: Affected family members Uses IBS ERPA: Extended Relative Pairs AnalysisDodgy statistic Genehunter NPL: Non-Parametric LinkageConservative Genehunter-PLUS: Likelihood (“tilting”) All these methods consider affected members only

Convergence of parametric and non-parametric methods  Curtis and Sham (1995) MFLINK: Treats penetrance as parameter Terwilliger et al (2000) Complex recombination fractions Parameters with no simple biological interpretation

Quantitative Sib Pair Linkage X, Y standardised to mean 0, variance 1 r = sib correlation V A = additive QTL variance (X-Y) 2 = 2(1-r) – 2V A (  -0.5) +  Haseman-Elston Regression (1972) Haseman-Elston Revisited (2000) XY = r + V A (  -0.5) + 

Improved Haseman-Elston  Sham and Purcell (2001)  Use as dependent variable Gives equivalent power to variance components model for sib pair data

Variance components linkage  Models trait values of pedigree members jointly  Assumes multivariate normality conditional on IBD  Covariance between relative pairs = Vr + V A [  -E(  )] WhereV = trait variance r = correlation (depends on relationship) V A = QTL additive variance E(  ) = expected proportion IBD

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

No linkage

Under linkage

Incomplete Marker Information  IBD sharing cannot be deduced from marker genotypes with certainty  Obtain probabilities of all possible IBD values Finite mixture likelihood Pi-hat likelihood

 QTL linkage model for sib-pair data P T1 QS N P T2 QSN 1 nqsnsq

Conditioning on Trait Values Usual test Conditional test Z i = IBD probability estimated from marker genotypes P i = IBD probability given relationship

QTL linkage: some problems  Sensitivity to marker misspecification of marker allele frequencies and positions  Sensitivity to non-normality / phenotypic selection  Heavy computational demand for large pedigrees or many marker loci  Sensitivity to marker genotype and relationship errors  Low power and poor localisation for minor QTL