{ } [ ] { } { } 1. The simple Wright-Fisher model. Mathematics Population Genetics. n-step transition probabilities eqn(11)

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1 Mhmcs Polo Gcs Corll Uvrsy J Jly 6 Wrr J ws Th sml Wrgh-shr modl q6 -s rso robbls q { } { }? r o r q { } { } { } { } } Prob{ δ δ δ δ δ δ { } [ ] / 4 gvs Ths. / - shr modl or h sml Wrgh δ δ M ms dffso romo q456 { } { } gros.8 } { log log log 4

2 M ms wh o l A g q34 shr Wrgh / log γ Codol rocss / / / q589 Codol m ms q3333 gvls q35 { } 4 gros { }.8 gros { } log gros λ Th Cgs modl q5 G lvs y dsrbo of y y y s dd of. As h Wrgh - shr modl ch g s hr of or A. offsrg gs. Th o lllc y A If Th h gvls of h mr { }r λ Prob{ X X λ y y y }.

3 3 q54 vr whr / y y y σ σ λ Th Mor b/d modl q5966 { } / / / M soor ms q64 / / M ms q6574 { } log log Codol m ms q { } / Lrgs o- gvl d s gvcors q7 / r λ

4 Th ldo-wly Modl O-wy mo: Wrgh-shr modl q36 "Mos of h m" hs bhvs s Mor modl. Howvr wh robbly chos o rrodc y m o lvs ψ w rmrs: γ dψ γ offsrg hs clds hmslf. h dvdl ψ ψ whrψ / O-wy mo: dffso romo q s h l frqcy of { } A { } M ms wh o-wy mo q7576 { } { } d d d Aohr rsso q4 Two-wy mo q ψ { v} µ v / v σ 4 v / / { v 4 4v } smll ordr rms Probwo gs of sm lllc y / 4

5 5 Homozygosy robbly q5 { }{ } / 4 Ifly my llls: Wrgh-shr modl q8 3 / d whr } Prob{ X Y X X Y Y Y Y π π π Homozygosy robbly q8384 { } { } Homozygosy robbly wh hr gs q8586 [ ] / Homozygosy robbly wh gs q88 [ ] / ml ro forml q93 Prob A

6 6 ml dsrbo of q vr / Prob Polo m of q35 φ d d q { } s fod bov / gs} Prob{oly o lll obsrvd sml of d Mor modl: h r olo q89 / Prob β β β β β β β β β β c Mor modl m mbr of llls wh rrsg gs q4 wh Comr hs m lll w - fo of qs Probbly of

7 7 / 3 / / olo llls h o : m mbr of wh Comr hs 3 Th cs M mbr of gros l loss of ll crr llls q3 q45 whr forml modl h olo c Mor h olds lll M g of w v w v Prors of h sml Wrgh- shr modl q [ ] { } [ ] { } [ ] Vr / / Vr Probwo gs hv sm r m m ν π π λ λ ffcv olo sz q / 4 modl Cgs Thrfor. / modl or h Cgs σ σ ffcv olo sz q34. or h Mor modl ν

8 8 gvl ffcv olo sz for h wo-gdr Wrgh- shr modl q44 4 gvl ffcv olo sz for h sb-dvdd olo Wrgh-shr modl q49 { } H H Ibrdg ffcv olo sz for h sb-dvdd olo Wrgh-shr modl q5 { } { } / H gvl ffcv olo sz for h cyclc olo sz Wrgh-shr modl q5 } { ml ro forml q93 Prob A ml dsrbo of q vr / Prob

9 9 Codol ro robbls q5 } Prob{ A ML of q54 ψ ψ q58 vr M ψ ψ q6 M τ 3 Th gm colsc Th gm colsc wh mos

10 Trcg bc o d sog mol vs Th gm colsc ssmos Wh hr r csrl ls gog bcwrds m Prob Prob colscc τ τ δτ mo τ τ δτ δτ δτ M m o MRCA of h sml q65 M g of olds lll h sml q68 T MRCA / gros. 4 gros Th gm colsc ssmos Wh hr r csrl ls gog bcwrds m Prob Prob colscc τ τ δτ mo τ τ δτ δτ δτ Wh s h glogy of dscr Mrov ch modl sffcly wll romd by h colsc? I rms of h Cgs modl wh σ Vr y σ f s s y < h sml 3 4 q

11 Th ldo-wly Modl Ths rqrms hold for h Wrgh - shr modl. vr y y < for ch 3 4 lso holds for ll.. "Mos of h m" hs bhvs s Mor modl. Howvr wh robbly chos o rrodc y m o lvs ψ w rmrs: γ dψ γ offsrg hs clds hmslf. h dvdl Th ldo - Wly modl bhvs h lm s h gm colsc rocss oly f γ >. Why? Thr s c colsc hory for h Mor modl. Ths ls for sml of y sz d lso for h r olo. Mor modl q7 q7 T MRCAP brh/dh vs vr T MRCA 4

12 Mor modl: m g of olds lll sml q74 v w whr v w Dsy fco of h frqcy of h olds lll h olo: q76 f m / M frqcy of h h olds lll h olo q Probbly h ll gs h sml r of h sm lllc y s h olds lll h olo: q Comr hs wh : d d Mor modl: robbly h h olds lll h olo s rrsd by gs q8 Mor modl M g of olds lll 4 gros. brh/dh vs q83

13 3 Dolly-Tvré-Grffhs forml: q86 Codol robbls of lll mbrs q9394 f Prob q96 9 A q vr g g g g q46 V D T T T < om mls of h hory

14 Th robbly h lll wh frqcy s h olds h olo by rvrsbly h robbly h lll wh frqcy srvvs h logs o h fr. Thrfor Prob mos frq s olds m frqcy of h mos frq. om mrcl vls:- Probbly Allls d vs. ss d. A lll s o rclr DA sqc. or ml w mgh hv:- A A s ggcgcggg.gcc s gcggggggg Vro c b cosdrd from h llls o of vw. or ml wh fv gs h s wh fv DA sqcs w mgh hv T G T C T G C T T G A T C T A T G C C T G C Vro c lso b cosdrd from h ss o of vw. Ths wh h sm fv gs DA sqcs w hv T G T C T G C T T G A T C T A T G C C T G C Ths gvs hr llls 3. β β β 3 Ths gvs for olymorhc gr ss s4 Ar w br off by sg llls h s horzol formo or by sg ss h s vrcl formo ssssg h dgr of olo vro d sg hyohss bo hs vro for ml wh hs csd? Vro s ssocd wh h olo gcs rmr 4. rom h llls vwo Prob wo DA sqcs r dffr /. rom h ss vwo h m of h mbr of ss whch wo DA sqcs dffr s. Boh r msrs of vro d boh r fcos of. 4

15 o w c rcs or qsos bo vro cocr wy by sg qsos bo. Do w bs sm hogh of s srrog for h mo of gc vro by sg llls d or by sg ss d? Covol wsdom swr s ss d. W r o bl o srvy h r olo d w hv o s d from sml of gs DA sqcs o sm. Allls d T G T C T G C T T G A T C T A T G C C T G C Thr llls 3. β β β 3 s sffc ssc for sg llls d Afr som clclo h m sqr rror of h smor of sg llls d s s d T G T C T G C T T G A T C T A T G C C T G C s d Th d cosss smly of h mbr of sgrgg ss h sml of gs DA qcs. g Vr g g whr g g or olymorhc ss s4 Ths s Vr s g d g g g 5

16 Vr om vls of M Why dos hs h? o: If ss r ld corrld Vr. g Ths s lwys lss h Vr Ths h roblm s gomrcl o of lg bw ss. A B Coml d wold dc h rm o whch vry mo occrrd. Allls d loss som formo bo h mbr of olymorhc ss for ml of h olymorhc s csd by h mo A. s d loss som formo bo wh mos occr. or ml f h mo C occrrd bw A d B hs formo s o rcordd. C Th llhood of sch d c b wr dow s fco of. Ths lds o h Crmér-Ro lowr bod for h vrc of y bsd smor of : Comrg h rsls fod by wo vsgors < { Th llls cs s smd sg h mbr of llls sml. Th rrow vrc of s whr s h mbr of llls s by vsgor smlg from h sm olo h sm gro boh sg sml of sz. 6

17 o h for wo rdom vrbls X d X hvg h sm robbly dsrbo Ε X X Vr X Cov X X Ths h rrow vrc s h brod vrc ms h covrc of d. r wh h cs. or gs r smld h frs wo by vsgor h fl wo by vsgor. Wh r h ossbls? Cofgro rs/cod Ivsgor AA/AA AA/BA AA/BB AB/AB AA/BC c. /3 AB/AC c. /3 AB/CD Cofgro 4 3 Probbly from [] Rqrd rrow vrc Ε o h ol vrc [ 8 3 ] rrow Vrc Tol Vrc

18 ml sz 3 gs r drw frs hr by vsgor fl hr by vsgor. or som cofgros - ms b. or ml hs s r of h cofgros {6}{} {}. ml: {} ABA CD c. - ABC DB c Prob {} D [A] [B] whr D of h 5 ossbls r of y [A] I ohr css - s somms zro somms o-zro. Corbo o Ε 3 4 D Collcg smlr rms Ε D Is hr br wy of fdg for sml szs 4 or mor? Ε Ths s mmzd wh.7 d h mmzg vl s.386 or hs vl of h brod vrc of s Ys. W s h vr d h bvr frqcy scr. Usg h frqcy scrm L h lll frqcs symoclly f olo b 3. Th f φ s O r Ε φ φ d ml. φ Ε[ ] d ml.φ Ε[ ] d Wr I I I 3. I I I 3. Whr I f olo lll s sml I ohrws. Th - I I [I I I I ] Ths Ε Ε I I Ε I I I I 8

19 I -I f d oly f lll A h olo s o sml o h ohr. Gv olo frqcs hs hs robbly - {-- }. Ths s h codol co of I -I gv h olo lll frqcs. Thrfor codolly Ε I I { } d... log or h rmg rm w d h bvr frqcy scrm: y y hvg h rory h Ε φ y y φ yddy ow Ε I I I I Ε I I Ε I I Gv. h cd vl of h ycl rms r Ths h cd vl of h dffrc of h ycl rms s Ths h codol vl of Ε I I I I y y [ y y ]dyd P y - o g [ ]dd [ d d] [ ]... s Ths gvs flly Ε... [ ]... Ths grs wh h rsls for 3 fod drcly. o: or lrg hs s romly log d s hs romly dd of. Ths romo s q ccr for >3. All of hs ssms << or mor ccrly smlg from f olo. Wh hs f hs s o so? Us smlo. Aohr bf: o cofrm h bov clclos. 9

20 mrcl Vrc Thorcl Vrc dcrsg o crsg o mrcl Vrc Avrg of wo sms Thorcl Vrc Icrsg o Icrsg o.39 om os. Th chg from h dcrsg o h crsg bhvor occrs romly wh.66.. Th horcl vrcs clcld bov ssm dfly lrg olo. Thy r ffc dffso romos. 3. As h sml sz rochs h mbr of gs h olo h mrcl vrc bcoms sbslly lss h h horcl vrc s cd. 4. Th smlg rsl rlll o M Vr [ψ '] smlg M log [ψ '] log [ ] s _~ log log 5. Th robbly of hrozygozy s ξ sy. Ths s smd sg by ξ ξ ξ Th smlg M of s romly log 4 log 6. Th robbly of hrozygozy c lso b smd by whr s h frqcy of lll h sml. Ths hs rom smlg vrc 4 3 wh dos hs m?

21 s d Rcll: If s h mbr of sgrgg ss sml of gs h g whr g / log Wr # sgrgg ss fod sml of gs by vsgor. Boh vsgors sml from h sm olo h sm gro. Wh s Ε? or lrg sy >3 hs s romly log. Ths h smlg vrc of h sm of wh w sm by o w sm by / g s / g s log _~ log g log comr wh fly my llls rsl.