Lecture 4 Resemblance Between Relatives

Transcription

1 Lecture 4 Resemblance Between Relatives Bruce Walsh jbwalsh@uariznaedu University f Arizna Ntes frm a shrt curse taught May 0 at University f Liege 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 V P = Variance f breeding values Phentypic Variance (4) Since an individual s phentype can be directly scred, the phentypic variance V P can be estimated frm measurements made directly n the randm breeding 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 knwn relatives) Thus estimates f V A require knwn 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 kinds f relatives is present Ancestral relatives: eg, parent and ffspring X X X k k k Measure phentypes f ne r bth parents, +k ffspring f each Cllateral relatives: Full Sibs have bth parents in cmmn X X X k k k Measure k ffspring in each family, but nt the parents Lecture 4, pg

2 Half Sibs have ne parent in cmmn n a a k k k k k k Measure phentype f k prgeny f each family, but nt the parents Nte that if k>, 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 the trait Further, if trait variatin has a significant genetic basis, the clser the relatives, the mre similar their appearance Phentypic Resemblance Between Relatives We nw will use the cvariance (and the related measures f crrelatins and regressin slpes) t quantify the phentypic resemblance between relatives Quantitative genetics as a field traces back t R A Fisher s 98 paper shwing hw t use the phentypic cvariances t estimate genetic variances, whereby the phentypic cvariance between relatives is 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 ( n ) Cv(O, P) b O P =, where Cv(O, P) = O i P i n O P Var(P) n where O i is the mean trait value in the ffspring f parent i and we examine n pairs f parentffspring 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 usually restricted t fathers The midparent-ffspring regressin slpe is estimated by b O MP = Cv(O, MP), where Cv(O, MP) = Var(MP) n i= ( n ) O i P MP,i no MP 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 Lecture 4, pg i=

3 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 estimated intraclass cvariances Under the simplest ANOVA framewrk, 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 value 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) (4) A key 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 result, let y ij = µ + b i + e ij be the jth member f grup i, where b i is the grup effect and e ij the residual, where σ(e ik,e ik )=0 The cvariance between tw members f grup i becmes σ(y ij,y ik )=σ(µ+b i +e ij,µ+b i +e ik )=σ(b i,b i )=σ (b) the between-grup variance (the variance in the grup effects) T see this pint, cnsider the fllwing extreme patterns f phentypes in full sib families: Situatin Lecture 4, pg

4 Here the between grup variance Var(B)=5, and the within-grup variance Var(W)=0 This gives a ttal phentypic variance f V P = 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 The resulting intraclass crrelatin t is t = Cv(full sibs) V P = Var(B) V P =09 where we have used the 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)=V P =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 phentypic resemblance is lw, s genetic variatin is lw Nte that phentypic resemblance amng relatives can equivalently be cnsider 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), as Cv(Within a grup) = Var(Between grup 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 may share similar envirnment The Genetic Cvariance Between Relatives The Genetic Cvariance Cv(G x,g y )= cvariance f the gentypic values (G x, G y ) f 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 the cvariances are estimated in practice Genetic cvariances arise because tw related individuals are mre likely 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 back t a single cpy in a recent cmmn ancestr Alleles can als be identical in state but nt identical by descent Lecture 4, pg 4

5 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 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 f 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 Likewise, the gentypic value f its ffspring is G = α + α + δ, giving Cv(G,G p )=Cv(α + α + δ,α +α +δ ) (4) We can use the rules f cvariances t expand this cvariance between tw sums int nine individual 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 The last identity fllws since Var(A)=Var(α +α )=Var(α ), s that (44) Var(α )=Cv(α,α )=Var(A)/ (45) Hence, when individuals share ne allele IBD, they share half the additive genetic variance Likewise, { 0 if xy wz, ie, bth alleles are nt IBD Cv(δ xy,δ wz )= (46) 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 (Equatins 44, 46), 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; Lecture 4, pg 5

6 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 What is the cvariance f gentypic values f tw full sibs? As illustrated previusly, 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) Lecture 4, pg 6

7 This results in a genetic cvariance between full sibs f Cv(G,G )= Var(A) + 4 (Var(A)+Var(D)) = Var(A) + Var(D) 4 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 taken t distinguish these envirnmental cvariances frm genetic cvariances 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 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 Hence, we can write E = E c + E g + E s, partitining the envirnmental variance as V E = V Ec + V Eg + V Es (47) 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 (48) 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 Cmplex Relatinships in Pedigrees: Cefficients f Cancestry Much f the analysis in animal breeding ccurs with pedigree data, where relatinships can be increasingly cmplex (ie, inbred relatives) We cnclude by intrducing the machinery t quantify such relatinships Lecture 4, pg 7

8 Cefficients f Cancestry and Inbreeding Suppse that single alleles are drawn randmly frm individuals x and y The prbability that these tw alleles are identical by descent, Θ xy, is called the cefficient f cancestry Anther way t lk at the prblem is t cnsider a hypthetical ffspring (z) f x and y By the abve definitin, Θ xy is the prbability that the tw genes at a lcus in individual z are identical by descent The latter quantity is Wright s (9) inbreeding cefficient, f z Thus, an individual s inbreeding cefficient is equivalent t its parents cefficient f cancestry, f z =Θ xy We nw prceed by example t demnstrate hw estimates f Θ xy are derived The first prblem t be tackled is the cefficient f cancestry f an individual with itself, Θ xx This may seem like a nnsensical task Hwever, we will sn see that Θ xx is an essential element f all cancestry estimates Dente the tw genes carried by individual x as A and A, and then randmly draw a gene frm the lcus, replace it, and randmly draw anther Θ xx is the prbability that the tw genes drawn are identical by descent There are fur ways, each with prbability /4, in which the genes can be drawn: A bth times, A first and A secnd, A first and A secnd, and A bth times If tw A genes are drawn, they must be identical by descent since they are cpies f the same gene The same applies t a draw f tw A genes Thus, prvided that genes A and A are nt identical t each ther by descent, then Θ xx is simply (/4)() + (/4)() = / We shuld, hwever, recgnize the pssibility that individual x is inbred, in which case the prbability that the gene A is identical by descent with the gene A is f x Thus, a general expressin fr the cefficient f cancestry f an individual with itself is Θ xx = 4 ( + f x + f x +)= ( + f x) (49) Maternal genes f p f p Offspring f Paternal gene Figure 4 The identity f genes by descent fr a parent and ffspring Circles and squares represent, respectively, maternally and paternally derived genes Left: The mther is nt inbred and her mate is nt a relative (s the ffspring is nt inbred) Center: The mther is inbred but unrelated t her mate Right: In additin t the mther being inbred, she is related t her mate, s that her ffspring is als inbred A slightly mre cmplicated situatin arises in calculating the cefficient f cancestry between a parent and its ffspring In rder t simplify the discussin, we will call the parent (p) f interest the mther, but the same results apply t fathers prvided the lcus is autsmal We first cnsider the situatin in which neither the mther nr her ffspring () are inbred, ie, the mther s parents are unrelated, and she is unrelated t her mate In that case, f the fur ways in which single genes can be drawn frm the mther and the child, nly ne invlves a pair that is identical by descent (Figure 4, left) Therefre, Θ p = /4 Suppse, hwever, that the mther is inbred (Figure 4, center), s that the prbability that bth f her alleles are identical by descent is f p This is the same as the prbability that the maternal gene inherited by the ffspring is identical by descent with the maternal gene nt inherited The prbability f drawing such a gene cmbinatin is /4 Therefre, inbreeding in the parent inflates Θ p t ( + f p )/4 With cmplete inbreeding (f p =), bth parental alleles are identical by descent, increasing Θ p t / Finally, we allw fr Lecture 4, pg 8

9 the pssibility that the parents f are related, s that the ffspring is inbred with cefficient f (Figure 4, right) It is nw necessary t cnsider the implicatins f drawing a paternally derived gene frm the ffspring, the prbability f which is / Since f is equivalent t the prbability that maternally and paternally derived genes are identical by descent, the additinal parent-ffspring identity induced by inbreeding is f / In summary, the mst general expressin fr the cefficient f cancestry fr a parent and ffspring is Θ p = 4 ( + f p +f ) (40) Often in the literature, Θ p is simply cnsidered t be /4 It shuld nw be clear that this implicitly assumes the absence f matings between relatives We nw mve n t the cefficient f cancestry f tw individuals that share the same father and mther (full sibs) We assume a species with separate sexes s that the mther and father are different individuals, and we again start with the simplest situatin, prgressively allwing the parents t be inbred and/r related (Figure 4) Fr the analysis f full sibs as well as mre cmplicated degrees f relatinship, the methd f path analysis (Lynch and Walsh Appendix ) prvides a useful tl The elements in Figure 4 n lnger represent gametes (as in Figure 4) but individuals / / ( + f m ) ( + f f ) ( + f m ) Θ ( + f f ) mf m f m f m f x y x y x y Figure 4 Path diagrams fr analyzing the prbability that randm genes frm tw full sibs are identical by descent The path cefficients alng single-headed arrws are always equal t / Left: The parents, m and f, are neither related nr inbred Center: The parents are unrelated, but inbred Right: In additin t being inbred, the parents are related with cefficient f cancestry Θ mf Let m represent the mther, f the father, and x and y their tw ffspring When the parents are neither inbred nr related, there are tw paths by which the same gene can be passed t bth x and y: x m y and x f y Since bth paths have identical cnsequences, we will simply cnsider the first f them First, we nte that the prbability that bth x and y receive the same maternal gene is / This is the cefficient f cancestry f the (nninbred) mther with herself, Θ mm, and is represented by the duble-headed arrw in the figure Secnd, we nte that the prbability f randmly drawing a maternal gene frm individual x is /, and that the same is true fr individual y Thus, the prbability f drawing tw maternal genes, identical by descent, ne frm x and the ther frm y, isθ mm /4=/8 Adding the same cntributin frm the paternal path, x f y, we btain the cefficient f cancestry Θ xy =/4 Path analysis prvides a simple way t btain this result First, set the path cefficients n all f the single-headed arrws in Figure 4 equal t / Then, nte that the cntributin f a path t a crrelatin between tw variables is equal t the prduct f the path cefficients and the crrelatin cefficient assciated with the cmmn factr (in this case, Θ mm r Θ ff =/) We nw allw fr the pssibility that the parents are inbred with inbreeding cefficients f m and f f, a cnditin that inflates the cefficient f cancestry f an individual with itself This is the nly necessary change fr the path diagram in Figure 4 (center) There are still nly tw paths that lead t genes identical by descent in x and y, and their sum is Lecture 4, pg 9

10 Θ xy = 4 (Θ mm +Θ ff )= ( +fm + +f ) f = 4 8 ( + f m + f f ) (4a) Finally, we allw fr the pssibility that m and f are related, such that the prbability f drawing tw genes (ne frm each f them) that are identical by descent is Θ mf It is then necessary t cnsider tw additinal paths between x and y: x m f y and x f m y (Figure 4, right) Again taking the cefficients n the single-headed arrws t be /, it can be seen that each f these tw new paths makes a cntributin Θ mf /4 t Θ xy Adding these t ur previus result, we btain a general expressin fr the cefficient f cancestry f full sibs, Θ xy = 8 ( + f m + f f +4Θ mf ) (4b) which reduces t Θ xy =/4under randm mating The preceding techniques are extended readily t mre distant relatinships and mre cmplicated schemes f relatedness The cefficient f cancestry is always the sum f a series f tw types f paths between x and y The first type f path leads frm a single cmmn ancestr t the tw individuals f interest, while the secnd type passes thrugh tw remte ancestrs that are related t each ther Neither type f path is allwed t pass thrugh the same ancestr mre than nce This prcedure is summarized by the fllwing equatin Θ xy = i Θ ii ( ) ni + j ( Θ jk j k ) njk (4) where n i is the number f individuals (including x and y) in the path leading frm cmmn ancestr i, and n jk is the number f individuals (including x and y) n the path leading frm tw different but related ancestrs, j and k Example 4 One f the first pedigrees t which Wright (9) applied his thery f inbreeding is that f Ran Gauntlet, an English bull In the fllwing figure, rectangles and vals refer t bulls and cws, respectively Lrd Raglan Mistlete Champin f England Duchess f Glster, 9th The Czar Mimulus Grand Duke f Glster Carmine Ryal Duke f Glster Princess Ryal Ran Gauntlet Lecture 4, pg 0

11 We wish t cmpute the cefficient f cancestry f the Ryal Duke f Glster and Princess Ryal This is the same as the inbreeding cefficient f their sn, Ran Gauntlet The fur pssible paths by which alleles identical by descent can be inherited by the Ryal Duke and Princess Ryal are indicated by the cded lines adjacent t the arrws in the pedigree Tw f these paths cntain fur individuals and tw cntain seven Thus, assuming that the remte ancestrs, Lrd Raglan and Champin f England, are nt inbred (s that fr bth, Θ ii =/), the cefficient f cancestry f the Ryal Duke and Princess Ryal is [(/) 4 +(/) 7 ]=04 This is a slightly clser relatinship than that fr half sibs (fr which Θ=05) Relative t the base ppulatin, the alleles at 4% f the autsmal lci in the ffspring, Ran Gauntlet, are expected t be identical by descent The Cefficient f Fraternity Up t nw we have been cnsidering the identity f single genes by descent Anther useful measure is the prbability that single-lcus gentypes (bth genes) f tw individuals are identical by descent The frmulatin f such a measure, which we dente as xy, is attributable t Ctterman (954) and was called the cefficient f fraternity by Trustrum (96) The prblem is set ut in Figure 4 Here we dente the mthers f individuals x and y as m x and m y, and the fathers as f x and f y The cefficients f cancestry Θ mxm y, Θ mxf y, Θ fxm y, and Θ fxf y prvide measures f the prbability f drawing genes identical by descent frm all fur cmbinatins f parents Θ fx m y Θ fx f y Θ mx m y f x m x Θ mx f y f y m y x y Figure 4 The analysis f the identity by descent f gentypes f individuals x and y f x and f y represent fathers (which may be the same individual) f x and y, respectively, whereas m x and m y represent their mthers Duble-headed arrws between tw parents represent cefficients f cancestry There are tw ways by which the gentype f x can be identical by descent with that f y: () the gene descending frm m x may be identical by descent with that descending frm m y, and that frm f x identical by descent with that frm f y, r () the gene frm m x may be identical by descent with that frm f y, and that frm f x identical by descent with that frm m y Thus, the cefficient f fraternity is defined as xy =Θ mxm y Θ fxf y +Θ mxf y Θ fxm y (4) Tw examples will suffice t illustrate the use f this equatin First, cnsider the situatin when x and y are full sibs, in which case the mthers are the same individual (m x = m y = m), as Lecture 4, pg

12 are the fathers f x = f y = f Equatin 4 then reduces t xy =Θ mm Θ ff +Θ mf (44) If the parents are unrelated, then Θ mf =0; and if the parents are nt inbred, then Θ mm =Θ ff =/ Substituting these values int the abve expressin, we btain xy =/4 Nw cnsider the case f paternal half sibs, in which case the fathers are the same individual, but the mthers are different Nw, xy =Θ mxm y Θ ff +Θ mxfθ fmy (45) Prvided that the parents are unrelated, then Θ ff =/and Θ mxm y =Θ mxf =Θ fmy =0, which yields xy =0 The gentypes f tw individuals cannt be identical by descent if their maternally (r paternally) derived genes cme frm unrelated individuals Example 4 Returning t the figure in Example 4, what is xy fr x = Ryal Duke f Glster and y = Princess Ryal? Designate the parents as f x = Grand Duke f Glster, m x = Mimulus, f y = Champin f England, and m y = Carmine Nting that Champin f England is the father f Grand Duke f Glster and Mimulus, Θ fxf y = Θ mxf y = (/4) Cunting the number f individuals in the paths f descent between the remaining tw pairs f parents, Θ mxm y =Θ fxm y =(/) 5 Substituting int Equatin 4, the prbability that x and y have identical gentypes by descent at an arbitrary autsmal lcus is xy =(/4)(/) 5 +(/) 5 (/4)=(/) 6 We nw have a cmplete system fr describing the identity by descent at an arbitrary lcus fr any tw individuals Fr cmplex pedigrees, this can be a rather tedius prcess, but relatively simple algrithms exist fr the cmputatin f Θ xy frm simple infrmatin n parentage (Lynch and Walsh Chapter 6) The identity cefficients fr several cmmn relatinships are summarized in Table 4 Genetic Crrelatins fr General Relatinships The abve results fr the cntributin when relatives share ne and tw alleles IBD suggests the general expressin fr the cvariance between (nninbred) relatives 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 (46) It can be esaily shwn that Θ xy = r xy, u xy = xy (47) We can thus rewrite Equatin 46 as Cv(G x,g y )=Θ xy V A + xy V D (48) Lecture 4, pg

13 If epistatic genetic variance is present, this can be generalized t Cv(G x,g y )=r xy V A + u xy V D + rxyv AA + r xy u xy V AD + u xyv DD + =Θ xy V A + xy V D +(Θ xy ) V AA +Θ xy xy V AD + xyv DD + (49) Table 4 Identity cefficients fr cmmn relatinships under the assumptin f n inbreeding, in which case t 6 =0 Relatinship Θ xy xy Parent ffspring Grandparent grandchild Great grandparent great grandchild Half sibs Full sibs, dizygtic twins Uncle(aunt) nephew(neice) First cusins Duble first cusins Secnd cusins Mnzygtic twins (clnemates) Lecture 4, pg

14 Lecture 4 Prblems Again cnsider the Brla Lcus (Lecture ) Suppse the envirnmental variance in litter size is 05 and there are n cmmn envirnmental effects due t maternal envirnment r rearing families tgether Assuming the Brla lcus is the nly surce f genetic variance, cmpute the regressins r intraclass crrelatins f litter size between the fllwing relatives: a: Offspring and midparent b: Half sibs c: Full sibs D the calculatins fr (i) ppulatins where freq(b) = 0 and (ii) ppulatins where freq(b) = 08 (Helpful hint might want t use results frm the Prblem set fr Lecture (prblems 4 and 5), unless yu want the extra practice!) What is the crrelatin between a grandparent and its grandchild? Lecture 4, pg 4

15 Slutins t Lecture 4 Prblems Recalling frm Prblem 5 frm the Lecture prblem set that : fr freq(b) =0:σ A =067, σ D=000, σ G=069 Hence σ P =0669 fr freq(b) =08:σ A =0090, σ D=000, σ G=009 Hence σ P =059 a) Midparent-ffspring regressin: b = V A /V p = 067/0669 = 050 fr freq(b) =0;05 fr freq(b) =08 b) Half-sib crrelatin: t HS =(/4)V A /V P =006 fr freq(b) =0;008 fr freq(b) =08 c) Full-sib crrelatin: t FS =(V A /+V D /4)/V P =06, fr freq(b) =0; 0077 fr freq(b) =08 A parent passes alng a single allele t its ffspring, and hence there is a / change that the ffspring passes n that allele t its ffspring Hence, Pr( alleles IBD) =0, while Pr( alleles IBD) = Pr(0 alleles IBD) =/ Hence, Cv(Grandparent, grandchild) = σ A /4 Lecture 4, pg 5