12-point linkage analysis10-810, CMB lecture 8---Eric XingModern genetics began with Mendel’s experiments on garden peas. He studied seven contrasting pairs of characters, including:The form of ripe seeds: round, wrinkledThe color of the seed albumen: yellow, greenThe length of the stem: long, shortMendel’s first law: Characters are controlled by pairs of genes which separate during the formation of the reproductive cells (meiosis)A aAaMendel’s two laws2Mendel’s second law: When two or more pairs of gene segregate simultaneously, they do so independently.A a; B bA B A b a B a bMendel’s two lawsMorgan’s fruitfly data (1909): 2,839 fliesEye color A: red a: purpleWing length B: normal b: vestigialAABB x aabbAaBb x aabbAaBb Aabb aaBb aabbExp 710 710 710 710Obs 1,339 151 154 1,195“Exceptions” to Mendel’s Second Law3Morgan’s explanationAAB Baabb×F1:AaB baabb×F2:AaB baabbAab baaBbCrossover has taken placeRecombinationParental types: AaBb, aabbRecombinants: Aabb, aaBbThe proportion of recombinants between the two genes (or characters) is called the recombination fraction between these two genes. It is usually denoted by r or θ. For Morgan’s traits:r = (151 + 154)/2839 = 0.107If 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.4One locus: founder probabilitiesFounders are individuals whose parents are not in the pedigree. They may of may not be typed. Either way, we need to assign probabilities to their actual or possible genotypes. This is usually done by assuming Hardy-Weinberg equilibrium. (There is a good story here.) If the frequency of D is .01, H-W says pr(Dd ) = 2x.01x.99Genotypes of founder couples are (usually) treated as independent.pr(pop Dd , mom dd ) = (2x.01x.99)x(.99)2D d1D d dd21Children get their genes from their parents’ genes, independently, according to Mendel’s laws; also independently for different children. D d D dd d321pr(kid 3 dd | pop 1 Dd & mom 2 Dd ) = 1/2 x 1/2One locus: transmission probabilities5pr(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 mutually exclusive and equiprobable ways 4 can get a D and a d.D dD dD dd dD D14532One locus: transmission probabilities - IIPedigree 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 independent of all other factors: genotypes at other loci, environment, genotypes and phenotypes of relatives, etc.Complete penetrance: pr(affected | DD ) = 1Incomplete penetrance: pr(affected | DD ) = .8DDDDOne locus: penetrance probabilities6One locus: penetrance - IIAge and sex-dependent penetrance (see liability classes)pr( affected | DD , male, 45 y.o. ) = .6D D (45)Assume penetrances pr(affected | dd ) = .1, pr(affected | Dd ) = .3 pr(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 together7Note that we begin by multiplying founder gene frequencies, followed by founder penetrances. Next we multiply transmission probabilities, followed by penetrance probabilities of offspring, using their independence given parental genotypes.If there are missing or incomplete data, we must sum over all mutually exclusive 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 in what is known as the Elston-Stewart algorithm for calculating probabilities over pedigrees. It is one of the two widely used approaches. The other is termed the Lander-Green algorithm and takes a quite different approach. Both are hidden Markov models, both have compute time/space limitations with multiple individuals/loci (see next) , and extending them beyond their current limits is the ongoing outstanding problem.One locus: putting it all together - IISon 3 produces sperm with D-T, D-t, d-T or d-t in proportions:21D 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 recombination8Son 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 1/2θ = 1/2 : independent assortment (cf Mendel) unlinked lociθ < 1/2 : linkedloci θ ˜ 0 : tightly linked loci Note: θ > 1/2 is never observedIf the loci are linked, then D-T and d-t are parental, and D-t and d-Tare recombinant haplotypes.Two loci: linkage and recombination -IIˆRecombination only discernible in the father. Here θ = 1/4 (why?)This is called the phase-known double backcross pedigree.D DT Td dt tD dt td dt tD dT tD dT tD dT td dt tTwo loci: estimation of recombination fractions9Two loci: phaseSuppose we have data on two linked loci as follows:Was the daughter’s D-T from her father a parental or recombinant combination? This is the problem of phase: did father get D-T from one parent and d-t from the other? If so, then the daughter's paternally derived haplotype is parental. If father got D-t from one parent and d-T from the other, these would be parental, and daughter's paternally derived haplotype would be recombinant.D dT td dt tD dT tTwo loci: dealing with phase• Phase is usually regarded as unknown genetic information, specifically, in parental origin of alleles at heterozygous loci.• Sometimes it can be inferred with certainty from genotype data on parents.• Often it can be inferred with high probability from genotype data on several children.• In general genotype data on relatives helps, but does not necessarily determine phase.• In practice, probabilities must be calculated under all phases compatible with the observed data, and added together. The need to do so is the main reason linkage analysis is computationally intensive, especially with multilocus analyses.10Two loci: founder probabilitiesTwo-locus founder probabilities are typically calculated assuming linkage equilibrium, i.e. independence of