Intrductin t Quantitative Genetics II: Resemblance Between Relatives Bruce Walsh 8 Nvember 006 EEB 600A The heritability f a trait, a central cncept in quantitative genetics, is the prprtin f variatin amng individuals in a ppulatin that is due t variatin in the additive genetic (ie, breeding) values f individuals: h = V A = Variance f breeding values Phentypic Variance Since an individual s phentype can be directly scred, the phentypic variance can be estimated frm measurements made directly n the ppulatin In cntrast, an individual s breeding value cannt be bserved directly, but rather must be inferred frm the mean value f its ffspring (r mre generally using the phentypic values f ther nwn relatives) Thus estimates f V A require nwn cllectins f relatives The mst cmmn situatins (which we fcus n here) are cmparisns between parents and their ffspring r cmparisns amng sibs We can classify relatives as either ancestral r cllateral, and we fcus here n designs with just ne type f relative In a mre general pedigree, infrmatin frm bth inds f relatives is present Ancestral relatives: eg, parent and ffspring X X X Measure phentypes f ne r bth parents, and ffspring Cllateral relatives: Full Sibs have bth parents in cmmn X X X Measure ffspring in each family, but nt the parents Intr t Quantitative Genetics, II
Half Sibs have ne parent in cmmn n a a Measure phentype f prgeny f each family, but nt the parents Nte that if >, this design invlves bth full- (within any clumn) and half-sibs (between clumns frm the same sire) and is referred t as a nested half-sib/full-sib design Key bservatin: The amunt f phentypic resemblance amng relatives fr the trait prvides an indicatin f the amunt f genetic variatin fr that trait If trait variatin has a significant genetic basis, the clser the relatives, the mre similar their appearance Phentypic Resemblance Between Relatives Quantitative genetics as a field traces bac t R A Fisher s 98 paper shwing hw the phentypic cvariance between relatives can be expressed in terms f genetic variances, as we detail belw Parent-ffspring regressins There are three types f parent-ffspring regressins: tw single parent - ffspring regressins (pltting ffspring mean versus either the trait value in their father P f r their mther P m ), and the midparent-ffspring regressin (the ffspring mean regressed n the mean f their parents, the midparent MP =(P f +P m )/) The slpe f the (single) parent-ffspring regressin is estimated by b O P = ( n ) Cv(O, P), where Cv(O, P) = O i P i n O P Var(P) n i= where O i is the mean trait value in the ffspring f parent i, O the ffspring mean ver all familes (als caled the grand mean f all ffspring), P the mean f all parents, and we examine n parent-ffspring families One culd cmpute separate regressins using males (P m ) and females (P f ), althugh the later ptentially includes maternal effect cntributins and hence single-parent regressins are usually restricted t fathers The midparent-ffspring regressin slpe is estimated by b O MP = ( n ) Cv(O, MP), where Cv(O, MP) = O i P mp,i n O MP Var(MP) n i= where O i is the mean trait value in the ffspring f parents in pair i, where these parents have an average trait value MP i and we examine n parent-ffspring pairs Ntice that all f the three regressins invlve the cvariance between parents and their ffspring Intr t Quantitative Genetics, II
Cllateral relatinships: ANOVA With cllateral relatives, the abve frmulae fr the sample cvariance is nt apprpriate, fr tw reasns First, there are usually mre than tw cllateral relatives per family Secnd, even if families cnsist f nly tw relatives, the rder f the tw is arbitrary ie, there is n natural distinctin between X and Y, as exists in the case f parents and ffspring Anther way f stating the secnd pint is that cllateral relatives belng t the same class r categry In cntrast, parents and ffspring belng t different classes The cvariance between parents and ffspring is an interclass (between-class) cvariance, while the cvariance between cllateral relatives is an intraclass (within-class) cvariance The analysis f variance (ANOVA), first prpsed in Fisher s 98 paper, is used t estimate intraclass cvariances Under the simplest ANOVA framewr, we can cnsider the ttal variance f a trait t cnsist f tw cmpnents: a between-grup (als called the amng-grup) cmpnent (fr example, differences in the mean values f different families) and a within-grup cmpnent (the variatin in trait value within each family) The ttal variance is the sum f the between and within grup variances, Var(T)=Var(B)+Var(W) () A ey feature f ANOVA is that the between-grup variance equals the within-grup cvariance Thus, the larger the cvariance between members f a family, the larger the fractin f ttal variatin that is attributed t differences between family means T see this pint, cnsider the fllwing extreme patterns f phentypes in full sib families: Situatin Suppse the between grup variance Var(B)=5, and the within-grup variance Var(W)=0 This gives a ttal phentypic variance f = Var(T)=Var(B)+Var(W)=7 Here: members f a family resemble each ther mre clsely than they d members f ther families there are large differences in average phentype between families Intr t Quantitative Genetics, II
The resulting intraclass crrelatin t is t = Cv(full sibs) = Var(B) =09 where we have used the abve-mentined ANOVA identity that the between-grup variance equals the within-grup cvariance (here, the cvariance between full sibs) Since elements f the same class are full-sibs, this is ften dented by t FS t distinguish it frm ther intraclass crrelatins Situatin Suppse the ttal (phentypic) variance is the same as in situatin, with Var(T)= =7 Hwever, suppse there is n between-grup variance (Var(B)=0), implying that Var(W)=7 and the intraclass crrelatin is t =0 Here: members f a family resemble each ther n mre than they d members f ther families there are n significant differences in average phentype between families (ie, all families have the same mean value) phentypic resemblance is lw, s genetic variatin is lw Nte that phentypic resemblance amng relatives can equivalently be cnsidered as a measure f the similiary amng a grup f relatives fr the phentype f a quantitative trait (the cvariance f family members), r the difference in phentype between different families (the between-grup variance = variance f family means) Causes f Phentypic Cvariance Amng Relatives Relatives resemble each ther fr quantitative traits mre than they d unrelated members f the ppulatin fr tw ptential reasns: relatives share genes The clser the relatinship, the higher the prprtin f shared genes relatives share the same envirnment The Genetic Cvariance Between Relatives The Genetic Cvariance Cv(G x,g y )= cvariance f the gentypic values (G x, G y ) f the related individuals x and y We will first shw hw the genetic cvariances between parent and ffspring, full sibs, and half sibs depend n the genetic variances V A and V D We will then discuss hw these cvariances are estimated in practice Genetic cvariances arise because tw related individuals are mre liely t share alleles than are tw unrelated individuals Sharing alleles means having alleles that are identical by descent (IBD): namely that bth cpies f an allele can be traced bac t a single cpy in a recent cmmn ancestr Alleles can als be identical in state but nt identical by descent Fr example, bth alleles in an A A individual are the same type (identical in state), but they are nly identical by descent if bth cpies trace bac t (descend frm) a single cpy in a recent ancestr Fr example, cnsider the ffspring f tw parents and label the fur allelic cpies in the parents by - 4, independent f whether r nt any are identical in state Intr t Quantitative Genetics, II 4
Parents: A A A A 4 Offspring: = A A = A A 4 = A A 4 = A A 4 Here, and share ne allele IBD, and share tw alleles IBD, and 4 share n alleles IBD Offspring and ne parent What is the cvariance f gentypic values between an ffspring (G ) and its parent (G p )? Denting the tw parental alleles at a given lcus by A A, since a parent and its ffspring share exactly ne allele, ne allele in the ffspring came frm the parent (say A ), while the ther ffspring allele (dented A ) came frm the ther parent T cnsider the genetic cntributins frm a parent t its ffspring, write the gentypic value f the parent as G p = A + D We can further decmpse this by cnsidering the cntributin frm each parental allele t the verall breeding value, with A = α + α, and we can write the gentypic value f the parent as G p = α + α + δ where δ dentes the dminance deviatin fr an A A gentype Liewise, the gentypic value f its ffspring is G = α + α + δ, giving Cv(G,G p )=Cv(α + α + δ,α +α +δ ) We can use the rules f cvariances t expand this int nine cvariance terms, Cv(G,G p )=Cv(α,α )+Cv(α,α )+Cv(α,δ ) + Cv(α,α )+Cv(α,α )+Cv(α,δ ) + Cv(δ,α )+Cv(δ,α )+Cv(δ,δ ) By the way have (intentinally) cnstructed α and δ, they are uncrrelated Further, { 0 if x y, ie, nt IBD Cv(α x,α y )= Var(A)/ if x = y, ie, IBD (a) The last identity fllws since Var(A)=Var(α +α )=Var(α ), s that Var(α )=Cv(α,α )=Var(A)/ Hence, when individuals share ne allele IBD, they share half the additive genetic variance Liewise, { 0 if xy wz, ie, bth alleles are nt IBD Cv(δ xy,δ wz )= (b) Var(D) if xy = wz, bth alleles are IBD Tw individuals nly share the dminance variance when they share bth alleles Using the abve identities ( a and b), eight f the abve nine cvariances are zer, leaving Cv(G,G p )=Cv(α,α )=Var(A)/ () Half-sibs Here, ne parent is shared, the ther is drawn at randm frm the ppulatin; Intr t Quantitative Genetics, II 5
The genetic cvariance between half-sibs is the cvariance f the genetic values between and T cmpute this, cnsider a single lcus First nte that and share either ne allele IBD (frm the father) r n alleles IBD (since the mthers are assumed unrelated, these sibs cannt share bth alleles IBD as they share n maternal alleles IBD) The prbability that and bth receive the same allele frm the male is ne-half (because whichever allele the male passes t, the prbability that he passes the same allele t is ne-half) In this case, the tw ffspring have ne allele IBD, and the cntributin t the genetic cvariance when this ccurs is Cv(α,α )=Var(A)/ When and share n alleles IBD, they have n genetic cvariance Summarizing: Case Prbability Cntributin and have 0 alleles IBD / 0 and have allele IBD / Var(A)/ giving the genetic cvariance between half sibs as Full-Sibs Bth parents are in cmmn, Cv(G,G )=Var(A)/4 (4) What is the cvariance f gentypic values f tw full sibs? Three cases are pssible when cnsidering pairs f full sibs: they can share either 0,, r alleles IBD Applying the same apprach as fr half sibs, if we can cmpute: ) the prbability f each case; and ) the cntributin t the genetic cvariance fr each case Each full sib receives ne paternal and ne maternal allele The prbability that each sib receives the same paternal allele is /, which is als the prbability each sib receives the same maternal allele Hence, Pr( alleles IBD) = Pr( paternal allele IBD) Pr( maternal allele IBD) = = 4 Pr(0 alleles IBD) = Pr( paternal allele nt IBD) Pr( maternal allele nt IBD) = = 4 Pr( allele IBD) = Pr( alleles IBD) Pr(0 alleles IBD) = We saw abve that when tw relatives share ne allele IBD, the cntributin t the genetic cvariance is Var(A)/ When tw relatives share bth alleles IBD, each has the same gentype at the lcus being cnsidered, and the cntributin is Cv(α + α + δ,α +α +δ )=Var(α +α +δ )=Var(A)+Var(D) Putting these results tgether gives Case Prbability Cntributin and have 0 alleles IBD /4 0 and have allele IBD / Var(A)/ and have allele IBD /4 Var(A)+Var(D) Intr t Quantitative Genetics, II 6
This results in a genetic cvariance between full sibs f Cv(G,G )= Var(A) + Var(A) (Var(A)+Var(D)) = 4 4 General degree f relatinship + Var(D) 4 Equatins a and b suggest a general expressin fr the cvariance between (nninbred) relatives, based n the prbabilities that they share ne and bth alleles IBD If r xy = (/) Prb(relatives x and y have ne allele IBD) + Prb(relatives x and y have bth alleles IBD), and u xy = Prb( relatives x and y have bth alleles IBD ), then the genetic cvariance between x and y is given by Cv(G x,g y )=r xy V A + u xy V D (6a) If epistatic genetic variance is present, this can be generalized t Cv(G x,g y )=r xy V A + u xy V D + r xyv AA + r xy u xy V AD + u xyv DD + (5) (6b) Envirnmental Causes f Relatinship Between Relatives Shared envirnmental effects (such as a cmmn maternal envirnment) als cntribute t the cvariance between relatives, and care must be taen t distinguish sharded envirnmental effects frm shared genetic effects If members f a family are reared tgether they share a cmmn envirnmental value, E c If the cmmn envirnmental circumstances are different fr each family, the variance due t cmmn envirnmental effects, V Ec, causes greater similarity amng members f a family, and greater differences amng families, than wuld be expected just frm the prprtin f genes they share Thus, V Ec inflates the phentypic cvariance f sibs ver what is expected frm their gentypic cvariance Just as we decmpsed the ttal gentypic value int cmpnents, sme shared, thers nt transmitted between relatives, we can d the same fr envirnmental effects In particular, we can write the ttal envirnmental effect E as the sum f a cmmn envirnmental effect shared by the relatives E c, a general envirnmental effect E g, and a specific envirnmental effect E s (unique t each indivdiual) Hence, we can write E = E c + E g + E s, partitining the envirnmental variance as V E = V Ec + V Eg + V Es (7) We can further cnsider different pssible surces f the cmmn envirnmental effect E c : E cs r E cl : Shared effects due t sharing the space/lcatin (different farms, cages) E ct : Tempral (changes in climactic r nutritinal cnditins ver time) E cm Maternal (pre- and pst-natal nutritin) Thus, we can partitin the envirnmental variance as V E = V Ec + V Eg + V Es = V EcS + V EcT + V EcM + V Ec + V Eg + V Es Cmmn envirnment effects mainly cntribute t resemblance f sibs, but maternal envirnment effects can cntribute t resemblance between mther and ffspring as well V EcS and V EcT can be eliminated, r estimated, by using the crrect experimental design, but it is very difficult (except by crss-fstering) t eliminate r estimate V EcM frm the cvariance f full sibs Further, crss-fstering nly remves pst-natal (past birth) maternal effects, it des nt remve shared pre-natal maternal effects Intr t Quantitative Genetics, II 7
Phentypic Cvariance Amng Relatives and h Summarizing the abve results, the resulting cvariances between cmmn sets f relatives, the assciated regressin slpes (b) r intra-class crrelatins (t), and hw these relate t estmates f h are as fllws: Relative Pair Cv t r b h Parent-ffspring (P -O) V A / b O P = V A/ b O P Midparent-ffspring (MP-O) V A / b O MP = V A / b O MP Half-sibs (HS) V A /4 t HS =(/4)V A / 4t HS Full-sibs (FS) V A /+V D /4+V Ec t FS = V A/+V D /4+V Ec t FS >h The midparent-ffspring slpe is cmputed as fllws: using the prperties f cvariances, Cv(O, MP) =Cv(0, [P f + P m ]/) = Cv(0,P f) = V A/ + V A/ =V A / + Cv(0,P m) The variance f the midparent values als fllws frm the prperties f cvariances, with ( ) Pf +P m Var(MP)=Var = Var(P f) 4 + Var(P m) 4 = / The last equality assumes equal trait variances in bth parents and that parental values are uncrrelated (ie, n assrtative mating) The regressin slpe equals the cvariance between midparent and ffspring divided by the midparent variance, b O MP = Cv(O, MP) Var(MP) = V A/ / = V A Intr t Quantitative Genetics, II 8
Resemblance Between Relatives Prblems The sm (small) allele is a bristle mutatin that segregates in an Australian Drsphila ppulatin, where the gentypic values fr the wildtype (+) and small (sm) alleles are as fllws: ++:+sm : sm sm have values f 44:40: Suppse the envirnmental variance f bristle number is 6, and there are n cmmn envirnmental effects due t maternal envirnment r rearing families tgether is vials Assuming the sm lcus is the nly surce f genetic variance, cmpute the regressins r intraclass crrelatins f bristle number between the fllwing relatives: a: Offspring and midparent b: Half sibs c: Full sibs D the calculatins fr (i) ppulatins where freq(sm) = 0 and (ii) ppulatins where freq(sm) = 09 Intr t Quantitative Genetics, II 9