An quick survey of human2-point linkage analysisStat 246, Lecture 2, Part BModern genetics began with Mendel’s experiments on gardenpeas. He studied seven contrasting pairs of characters,including: The form of ripe seeds: round, wrinkled The color of the seed albumen: yellow, green The length of the stem: long, shortMendel’s first law: Characters are controlled by pairs of geneswhich separate during the formation of the reproductive cells(meiosis)A aAaMendel’s two lawsMendel’s second lawWhen two or more pairs of genes segregatesimultaneously, they do so independently.A a; B bA B A b a B a b“Exceptions” to Mendel’s Second LawMorgan’s fruitfly data (1909): 2,839 fliesEye color A: red a: purpleWing length B: normal b: vestigialAABB x aabbAaBb x aabb AaBb Aabb aaBb aabbExp 710 710 710 710Obs 1,339 151 154 1,195Morgan’s explanationAAB Baabb×F1:AaB baabb×F2:AaB baabbAab baaBbCrossover has taken placeParental types: AaBb, aabbRecombinants: Aabb, aaBbThe proportion of recombinants between the two genes(or characters) is called the recombination fraction betweenthese two genes. It is usually denoted by r or θ. For Morgan’s traits: r = (151 + 154)/2839 = 0.107 If r < 1/2: two genes are said to be linked. If r = 1/2: independent segregation (Mendel’s second law). Now we move on to (small) pedigrees.One locus: founder probabilitiesFounders are individuals whose parents are not in the pedigree. They may ofmay not be typed. Either way, we need to assign probabilities to their actualor possible genotypes. This is usually done by assuming Hardy-Weinbergequilibrium. (There is a good story here.) If the frequency of D is .01, H-Wsays pr(Dd ) = 2x.01x.99Genotypes of founder couples are (usually) treated as independent. pr(pop Dd , mom dd ) = (2x.01x.99)x(.99)2D d D d dd121One locus: transmission probabilitiesChildren get their genes from their parents’ genes,independently, according to Mendel’s laws; alsoindependently for different children.D d D dd d321pr(kid 3 dd | pop 1 Dd & mom 2 Dd )= 1/2 x 1/2One locus: transmission probabilities - IID dD dD dpr(3 dd & 4 Dd & 5 DD | 1 Dd & 2 Dd ) = (1/2 x 1/2)x(2 x 1/2 x 1/2) x (1/2 x 1/2).The factor 2 comes from summing over the two mutuallyexclusive and equiprobable ways 4 can get a D and a d.d dD D14532One locus: penetrance probabilitiesPedigree analyses usually suppose that, given the genotype at all loci,and in some cases age and sex, the chance of having a particularphenotype depends only on genotype at one locus, and is independentof all other factors: genotypes at other loci, environment, genotypes andphenotypes of relatives, etc.Complete penetrance:pr(affected | DD ) = 1Incomplete penetrance:pr(affected | DD ) = .8DDDDOne locus: penetrance - IIAge and sex-dependent penetrance (see liabilityclasses) pr( affected | DD , male, 45 y.o. ) = .6D D (45)One locus: putting it all togetherAssume penetrances pr(affected | dd ) = .1, pr(affected | Dd ) = .3pr(affected | DD ) = .8, and that allele D has frequency .01.The probability of this pedigree is the product:(2 x .01 x .99 x .7) x (2 x .01 x .99 x .3) x (1/2 x 1/2 x .9) x (2 x 1/2 x 1/2 x.7) x (1/2 x 1/2 x .8)D d D dD dd dD D14532In general shaded means affected, blank means unaffected.One locus: putting it all together - IINote that we begin by multiplying founder gene frequencies, followedby founder penetrances. Next we multiply transmission probabilities,followed by penetrance probabilities of offspring, using theirindependence given parental genotypes.If there are missing or incomplete data, we must sum over all mutuallyexclusive possibilities compatible with the observed data.The general strategy of beginning with founders, then non-founders,and multiplying and summing as appropriate, has been codified inwhat is known as the Elston-Stewart algorithm for calculatingprobabilities over pedigrees. It is one of the two widely usedapproaches. The other is termed the Lander-Green algorithm andtakes a quite different approach.Both are hidden Markov models, both have compute time/spacelimitations with multiple individuals/loci (see next) , and extendingthem beyond their current limits is the ongoing outstanding problem.Two loci: linkage and recombinationSon 3 produces sperm with D-T, D-t, d-T or d-t inproportions:321D dT td dt tD DT T3T tD (1-θ)/2 θ/2 1/2d θ/2 (1-θ)/2 1/21/2 1/2Two loci: linkage and recombination - IISon produces sperm with DT, Dt, dT or dt in proportions:T tD (1-θ)/2 θ/2 1/2d θ/2 (1-θ)/2 1/21/2 1/2 θ = 1/2 : independent assortment (cf Mendel) unlinked loci θ < 1/2 : linked loci θ ≈ 0 : tightly linked lociNote: θ > 1/2 is never observedIf the loci are linked, then D-T and d-t are parental, andD-t and d-T are recombinant haplotypesˆ Recombination only discernible in the father. Here θ = 1/4 (why?) This is called the phase-known double backcross pedigree.Two loci: estimation of recombinationfractionsD DT Td dt tD dt td dt tD dT tD dT tD dT td dt tTwo loci: phaseD dT td dt tD dT t Suppose we have data on two linked loci as follows: Was the daughter’s D-T from her father a parental or recombinantcombination? This is the problem of phase: did father get D-T from oneparent and d-t from the other? If so, then the daughter's paternallyderived haplotype is parental. If father got D-t from one parent and d-T from the other, these wouldbe parental, and daughter's paternally derived haplotype would berecombinant.Two loci: dealing with phasePhase is incompleteness in genetic information, specifically, in parentalorigin of alleles at heterozygous loci.Often it can be inferred with certainty from genotype data on parents.Often it can be inferred with high probability from genotype data onseveral children.In general genotype data on relatives helps, but does not necessarilydetermine phase.In practice, probabilities must be calculated under all phases compatiblewith the observed data, and added together. The need to do so is themain reason linkage analysis is computationally intensive, especiallywith multilocus analyses.Two loci: founder probabilities Two-locus founder probabilities are typically calculated assuminglinkage equilibrium, i.e.
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