GenCB 511 Coarse Notes Population Genetics NONRANDOM MATING & GENETIC DRIFT

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1 NONRANDOM MATING & GENETIC DRIFT NONRANDOM MATING/INBREEDING READING: Hartl & Clark, Wll dstngush two tyes of nonrandom matng: (1) Assortatve matng: matng between ndvduals wth smlar henotyes or among ndvduals that occur n a artcular locaton. () Inbreedng: matng between related ndvduals. Both tyes of nonrandom matng may have smlar consequences snce ndvduals wth smlar henotyes often have smlar genotyes. It s often dffcult to searate cause from effect. E.g., ndvduals wth smlar henotyes may mate because a) henotyc assortatve matng occurs; b) matng wth relatves s referred; c) matngs are rmarly based on roxmty. Poulaton subdvson: The Wahlund Effect It turns out that oulaton subdvson er se can effect the dstrbuton of genotyes n the entre oulaton. Consder a locus wth alleles (A and a) and a collecton of solated suboulatons numbered 1,, 3,... Let the frequences of A and a n suboulaton be and q. Assumng random matng wthn each (solated) suboulaton: Freq(AA) n suboulaton Freq(Aa) n suboulaton q Freq(aa) n suboulaton q Let Avg ( ) average freq. of A across all suboulatons. ( ) 1 - Lkewse, let q Avg q What s the average frequency of each genotye over all the suboulatons? Consder AA homozygotes frst: From Fun Facts: Var( X) E X So, Avg ( ) ( ) [ E( X) ] so E( X ) [ E( X) ] + Var X ( ) + Var ( ) II-1

2 Smlarly, for aa homozygotes: Avg q ( ) q q + Var q Fnally, for Aa heterozygotes: Avg q A Thought Exerment: ( ) q + Var( ) snce Var(q) Var(1 ) Var(). ( ) 1 q 1 q Var ( ) q Var( ). Suose genotyes were randomly samled from a oulaton whose substructure was unknown. The frequences of A and a n samle would be and q. Wth random matng, would then exect to fnd genotyes n roortons AA : Aa :aa : q : q. But, the genotye frequences observed would be AA : Aa :aa : q : q + Var( ): q Var( ) : q + Var( ). I.e., would fnd an excess of homozygotes and a defct of heterozygotes, comared to exectatons. Why? Smly because of oulaton subdvson and, n artcular, varance n allele frequences across suboulatons. Gven across-suboulatons dfferences n allele frequences, the aarent excess n homozygotes and defct of heterozygotes from what s exected were the entre oulaton to mate at random defnes what s called the Wahlund Effect. The Wahlund effect s a common cause of non-conformty to Hardy-Wenberg exectatons n oulaton samles. INBREEDING Wll now consder the genetc consequences of matng between relatves: nbreedng. In 190's, Sewall Wrght nvented an ngenous aroach to trackng genotyes through edgrees based on the robablty that allele coes are dentcal by descent. Wll follow the French genetcst Malecot's reworkng of Wrght's method here. e.g., B C D E G II-

3 Identty by descent (IBD): Two alleles are dentcal by descent f (1) both are descended from the same allele n a common ancestor or () one allele s descended from the other. Wll mean IBD relatve to a secfc base oulaton (whose alleles are deemed to be not IBD). Defnton: The nbreedng coeffcent, f, of an ndvdual s the robablty that ts two gene coes at a locus are dentcal by descent. Once f s known, t's not hard to fnd the robabltes that s AA, Aa, or aa: Consder a randomly chosen ndvdual: Wth robablty f, both gene coes n that ndvdual are IBD. Then both wll be A f the allele they were coed from n the base oulaton were A. But, A occurs wth frequency n the base oulaton, so the robablty of beng an AA gven both genes are IBD s. the robablty of gettng two A alleles that are IBD s f. Lkewse, the robablty of beng aa wth both genes are IBD s f ( 1 ). Wth robablty 1 f, the two genes n an ndvdual wll not be IBD. Must have descended from dfferent allele coes n the base oulaton. Assumng the coes are made ndeendently, then wth robablty, the coed alleles are both A (genotye AA), etc. Puttng ths all together, have: ( 1 f ) + f P Aa ( 1 f ) ( 1 ) ( )( 1 ) + f ( 1 ) P aa 1 f Great! So now only need to determne f. Thanks to Wrght, ths s very easy to do. C E.g., Let's fnd f n the edgree above. c c' Need only concentrate on the central art of the edgree: f Prob e c ( ) Prob( c c ) Prob ( c g) E G e g II-3

4 So the exected genotye frequences for ths edgree are: ( 7 8) P Aa ( 7 8)( 1 ) 7 8 ( )( 1 ) + 1 8( 1 ) ASIDE: What s the average frequency of A among ndvduals wth nbreedng coeffcent f? Freq A [ ] + 1 [( 1 f ) ( 1 ) ] [ ] + f ( 1 f ) + f ( ) + 1 P 1 f Aa ( ) + f ( 1 f ) + Inbreedng does not affect allele frequences on average, but does affect the robabltes that A's or a's co-occur n an ndvdual. Comutng Inbreedng Coeffcents (n general) Say we want to fnd f I n the edgree at rght: Rules: B (1) Enumerate each loo () Each loo must C D a) go through each ndvdual no more than once b) only change from u to down once (3) Multly by 1 for each assage through an ndvdual If the assage through an ndvdual nvolves a change of drecton (u/down) multly by 1 + f ndvdual. ( ) nstead of 1/, where f s the nbreedng coeffcent for that (4) Add the robabltes of each loo. For the above examle: f I G E C G E D B ( ) 1 D 1 E G ( ) 1 E ( ) 1 ( ) G 1+ fe I II-4

5 Some forbdden loos: IGECBDEGI (goes through G twce) IDBCEGI(already counted) IGECBDEI (loo ECBDE already accounted for n f E ) Evolutonary Alcaton: Kn Selecton Probablty that two ndvduals share an allele descended from a common ancestor s called the knsh coeffcent or coeffcent of consangunty. Knsh coeffcent between ndvduals A and B s denoted F AB. What s F G n the last examle? Clearly, t s the nbreedng coeffcent of ther offsrng I, f I 7 3. The connecton between f and F: The knsh coeffcent of two ndvduals s equal to the nbreedng coeffcent of ther (erhas hyothetcal) offsrng e.g., F mother,daughter 1/4 (assumng unrelated, non-nbred arents). Knsh coeffcents useful n studyng evoluton of socal (esecally "altrustc") trats. In 1964, W. D. Hamlton roosed a rule-of-thumb to determne whether a rare allele wll be favored by selecton: A rare mutaton whch affects the ftness of ts carrer and others wll sread f br > c where b "beneft" ncrease n ftness to recents of acton c "cost" loss n ftness to actor r "degree of relatedness". can use above rules to fnd r: r F Another vew of f : f s the roorton by whch heterozygosty s reduced relatve to a random matng grou wth the same allele frequences. If f 0, the oulaton s n Hardy-Wenberg roortons. If f 1, all ndvduals are homozygous. In general: Het f ( 1 f )Het 0 where Het 0 ( 1 ) q Consder, e.g., the Wahlund effect: Overall frequences of A and a are and q. If ndvduals from all suboulatons mate at random, exect to fnd q heterozygotes. Would actually observe q Var( ) heterozygotes. Thus, n ths case q Var( ) ( 1 f ) q. In order for ths to be true, f Var( ) q. [Suggested exercse: show ths.] Note: f 0 even though there s no defct of heterozygotes wthn suboulatons and no obvous nbreedng! II-5