Recall that quantitative genetics is based on the extension of Mendelian principles to polygenic traits.

Transcription

1 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Readng: Chapter 5 and 6. Extensons to Multlocus trats Recall that quanttatve genetcs s based on the extenson of Mendelan prncples to polygenc trats. Untl now we modeled our expected phenotype assumng a sngle loc nfluences the trat. We had two types of effect: addtve and domnance. To extend our model to nclude multlocus trats, we need to consder not only the effects correspondng to each locus but also the effects correspondng to nteractons between loc. Epstass: an example There are two loc n mce that correspond to two color genes that affect the pgment granules n mouse coat har (example from Falconer and Mackay 1996). Locus Locus 1 B- bb C cc Table.1: Mean number of melann granules per unt volume of har. B and C are domnant alleles for the two loc. Locus Locus 1 B- bb C cc Table.: Mean sze of melann granules. B and C are domnant alleles for the two loc. 1

2 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Epstass: the model for two loc Consder two loc such that an ndvdual has alleles and at the frst locus and alleles B k and B l at the second locus. Let kl represent the genotypc value for ths ndvdual. More rgorously, kl The basc model decomposes kl nto a component due to addtvty between loc and a component due to devaton from ths addtvty (sound famlar?) kl µ + ( α + α + δ ) + ( α + α + δ ) + ε k l kl kl Here, ε kl s the devaton from addtvty of the effects for the two loc. Ths could correspond to any of the possble types of nteracton: addtve addtve, addtve domnance or domnance domnance. It s straghtforward to extend the model to three loc: klmn There are then more types of possble nteracton: addtve addtve, addtve domnance, domnance domnance, addtve addtve addtve, addtve addtve domnance, addtve domnance domnance, domnance domnance domnance

3 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout eneral Least-Squares Model for enetc Effects We have the followng model: Here, ε kl kl µ + ( α + α + δ ) + ( α + α + δ ) + ε represents all nteractons between loc. If we expand ε kl k l kl kl, then our model becomes kl µ + ( α + α + δ ) + ( α + α + δ ) + ( αα ) k l kl k + ( αα ) l + ( αα ) k + ( αα ) l + ( αδ ) kl + ( αδ ) kl + ( αδ ) k + ( αδ ) l + ( δδ ) kl We can then proceed as before usng least-squares to derve formulae for each component. Denote the condtonal mean genotypc value for ndvduals wth allele at the frst locus by... and smlarly let.k. denote the condtonal mean genotypc value for ndvduals wth allele at the frst locus and allele k at the second locus and so forth for other condtonal mean genotypc values. The addtve effect for allele s defned as α The domnance effect for genotype of locus one s δ Hgher nteractons are defned as follows: (αα) k 3

4 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout (αδ) kl (δδ) kl Thus we have parttoned the genotypc value begnnng wth the lowest order terms, accountng for as much varaton as possble. Ths contnues dealng wth progressvely more complex nteractons. ven that there s random matng and that the loc are ndependent (unlnked), there s no statstcal relatonshp between the genes found wthn or among loc. Why? Whch assumpton gves us the wthn ndependence? the among ndependence? Usng ths fact, the total genetc varance s the sum of the varance of the ndvdual effects: D D DD 4

5 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Example 5.1 of the Text contans measurements of the average length of vegetatve nternodes n the lateral branch for a corn varety. Measurements are gven by genotype class for two dallelc loc. BV30 U M U M U M U T U T U T a k B M B M B M B T B T B T a k The dfferent varance components are calculated to be: D D DD Note that / of the varaton s accounted for by the addtve effects of alleles. Thngs to keep n mnd: Lnkage Dsequlbrum We have dscussed the possblty of dependence between alleles wthn a locus (Hardy- Wenberg Dsequlbrum). dependence can exst for alleles between loc as well. We wll begn by consderng gamete frequences for two dallelc loc: 5

6 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout If the allele at one locus occurs ndependently of the allele at the other locus, then we would expect: However, dependence can be caused by many factors such as natural selecton, founder effects, mgraton, mutaton, random drft etc. measure of ths dependence s the lnkage dsequlbrum coeffcent or the coeffcent of gametc phase dsequlbrum: The relatonshp between D B and c B Example of lnkage dsequlbrum: (oka et al. 1997) Hemochromatoss s an autosomal recessve dsorder that results n a buld-up of ron. The responsble locus has been mapped to 6p. From OMIM: The features of hemochromatoss nclude crrhoss of the lver, dabetes, hypermelanotc pgmentaton of the skn, and heart falure. Prmary hepatocellular carcnoma (HCC; ), complcatng crrhoss, s responsble for about one-thrd of deaths n affected homozygotes. Snce hemochromatoss s a relatvely easly treated dsorder f dagnosed, ths s a form of preventable cancer. oka and collegues studed dsequlbrum for 4 polymorphsms n an 8 Mb regon of 6p. The followng plot shows the absolute value of the lnkage dsequlbrum coeffcent for all pars of these loc: 6

7 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout For addtonal nformaton on measures of dsequlbrum see Wer BS (1996) enetc Data nalyss II. There are tons of tests of dsequlbrum. Try a Medlne search for lnkage dsequlbrum or genetc assocaton or gametc phase dsequlbrum or transmsson dsequlbrum test or any smlar combnaton. We wll come back to such tests but for now we need to note that the presence of dsequlbrum wll affect our addtve and domnance varances. Consder n dallelc loc: n n n α( ) pq α( ) α( ) D α() represents the average effect of allelc substtuton at the th locus. D 4 n n n ( ak pq ) a a k k D 7

8 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Estmaton of D B ssumng random matng, an unbased estmator of D B can be found by Dˆ B N 4N BB + ( N Bb + N abb ) + N abb pˆ pˆ B N 1 N wth an estmate of varance Var ˆ ( Dˆ ) pˆ pˆ a pˆ B N 1 pˆ b ( pˆ + 1)( pˆ N B 1) Dˆ + ˆ D N( N 1) Where are we n the grand scheme of thngs? We can descrbe varaton n a trat as a functon of a multlocus genotype. Ths ncludes beng able to detect non-random assortment of alleles at a sngle locus detect non-random assortment of alleles between two loc descrbe the genotypc effects for a sngle locus descrbe the nteracton between genetc nformaton at multple loc derve the varance correspondng to the addtve effects of alleles, domnance effects, and nteracton terms 8

9 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Sources of Envronmental Varaton eneral envronmental effects - nfluental factors that are shared by groups of ndvduals Specal envronmental effects - resdual devatons from the phenotype expected on the bass of genotype and general envronmental effects enotype envronment nteracton the process by whch genotypes respond to envronmental change n dfferent ways Extenson of lnear model: z + I + E + e k k for ndvdual k wth genotype n envronment. µ µ E + + I + E e k We now can decompose the phenotypc varaton nto components for genotype, envronment (both general and specfc), gene envronment nteracton and genotype-envronment covarance: P I, E E e 9

10 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout Example: Fnd, E, I and µ for the followng data. ssume that there are an equal number of ndvduals n each genotype-envronment class. Envronment enotype class E µ I 11 I 1 I 1 I I 31 I 3 e k Interpretaton Suppose that the three genotypc classes are defned by a dallelc locus wth classes 1, and 3 correspondng to genotypes, a and aa, respectvely. Further, suppose that ths locus has a maor gene affectng our trat of nterest. In other words, all varaton due to other genetc causes s neglgble. nswer the followng questons. Suppose you are unaware of the envronmental effect. What would be our three mean genotypc values? How would you descrbe the mode of nhertance? 10

11 BIOSTT/STT551, Statstcal enetcs II: Quanttatve Trats Wnter 004 Sources of varaton for multlocus trats and Handout If hgh trat values (>35) are detrmental and demand an ndvdual undergo nvasve treatment, what would you recommend to a heterozygous ndvdual? How would knowledge of the envronmental factor nfluence ths advce? Now t has come to your attenton that there s ndeed an envronmental effect. Further, the exposure to envronment one and the presence of an allele ncrease an ndvdual s level of the trat. Is ths relatonshp purely addtve? How do you support your answer? Suppose I tell you that the locus doesn t correspond to a functonal gene but a marker n a non-codng regon of the chromosome. ve an explanaton for why we could stll see ths relatonshp. What step mght you take next? Suppose you were only gven nformaton on the envronmental varable and the trat. If the only other contrbuton to varaton n our trat was random (normally dstrbuted) nose, how would the overall trat dstrbuton look? How would the trat dstrbuton look wth-n each envronmental class? 11