Shortcut predictions for fitness properties at the MSD balance and for its. build-up after size reduction under different management strategies.

Transcription

1 Geneics: Published Aricles Ahead of Prin, published on April 5, 2007 as 0.534/geneics Shorcu predicions for finess properies a he MSD balance and for is build-up afer size reducion under differen managemen sraegies. Aurora García-Dorado Deparameno de Genéica. Faculad de Biología. Universidad Compluense Madrid. Spain

2 Running head: Managing he shif o a new MSD balance Keywords: inbreeding depression, addiive variance, dominan variance, purging, equal family conribuion, geneic managemen, conservaion. Corresponding auhor: Aurora García-Dorado. Deparameno de Genéica. Faculad de Biología. Universidad Compluense Madrid. Spain. Tf : Fax: augardo@bio.ucm.es 2

3 Absrac For populaions a he muaion-selecion-drif (MSD) balance, I develop approximae analyical expressions giving expecaions for he number of deleerious alleles per gamee, he number of loci a which any individual is homozygous for deleerious alleles, he inbreeding depression rae and he addiive and dominan componens of finess variance. These predicions are compared o diffusion ones, showing good agreemen under a wide range of siuaions. I also give approximaed analyical predicions for he changes in mean and addiive variance for finess when a populaion approaches a new equilibrium afer is effecive size is reduced o a sable value. Resuls are derived for populaions mainained wih equal family conribuion or wih no managemen afer size reducion, when selecion acs hrough viabiliy or feriliy differences. Predicions are compared o previously published resuls obained from ransiion marices or sochasic simulaions, a good qualiaive fi being obained. Predicions are also obained for populaions of various sizes under differen ses of plausible muaional parameers. They are compared o available empirical resuls for Drosophila, and conservaion implicaions are discussed. 3

4 The evoluionary fae of species and he survival prospecs of endangered populaions depends, among oher facors, upon he geneic changes in average finess occurring hrough generaions. Of paricular imporance for conservaion is he behaviour of finess in populaions undergoing a subsanial reducion in census size. For a given populaion size and rae of inpu of deleerious muaion, he geneic change for finess depends upon properies such as he inbreeding depression rae and genoypic componens of variance for finess. For populaions a he balance beween deleerious muaion, selecion and drif, expecaions for hese geneic properies can be accuraely prediced by means of he diffusion mehod (Kimura, 969), which has proven o be a powerful and reliable ool hrough exensive simulaion sudies. Diffusion predicions have been obained (García-Dorado 2003) using differen plausible experimenal esimaes of muaional parameers (see García-Dorado e al for a review). However, he diffusion mehod usually does no provide racable analyical expressions for equilibrium finess geneic properies o be used in furher predicions of finess evoluion, bu requires numerical compuaion o arrive a a predicion. Finess changes afer a sable reducion in populaion number can be prediced using he ransiion marix mehod (Lynch e al. 995), bu his is compuaionally very cosly, and does no provide analyical expressions wih heurisic informaion on he process. Here I develop approximae analyical expressions ( overall frequency approximaions) for he geneic properies of finess in populaions a he muaion-selecion-drif balance and for he behaviour of he mean and addiive variance for finess during he populaion s shif from an ancesral equilibrium o a new one for a smaller effecive number. A common sraegy used o increase he effecive size (N e ) during ex siu conservaion of small endangered populaions consiss of equaing family conribuions o he breeding pool (EC mehod), rendering N e values abou wice he acual populaion 4

5 number (Wrigh 938). This sraegy has differen implicaions regarding he geneic diversiy mainained or he adapaion o capive condiions (Frankham e al. 2002), bu here I will only deal wih is impac on he rae of change in geneic properies of finess assuming ha genoypic finess values during he conservaion process are he same as in he ancesral populaion. In his respec, EC reduces he per generaion inbreeding rae, bu also avoids naural selecion for fecundiy and reduces viabiliy selecion o ha due o wihin-family geneic variabiliy, hus allowing increased accumulaion of deleerious muaions (Fernández and Caballero, 200). To assess he efficiency of EC in differen insances, he finess performance under EC mus be compared o ha obained wih no managemen (NM), when family conribuions randomly depend on family feriliy and offspring viabiliy. This has been achieved for some specific cases using compuer simulaion and ransiion marix predicions (Schoen e al. 998, Fernández and Caballero 200, Theodorou and Couve 2003) or carrying ou Drosophila experimens (Borlase e al. 993, Rodríguez-Ramilo e al. 2006) bu, so far, no general approach has been advanced o obain analyical predicions for he relaive advanage of EC and NM mehods. I have adaped he above shif approximaion o he case where EC is applied, eiher considering selecion operaing on fecundiy or viabiliy, and I use hese approximaions o infer he condiions for which he EC mehod of mainenance is superior under various ses of muaional parameers. ANALYTICAL RESULTS The deerminisic model Assume a panmicic diploid populaion where non-recurren deleerious muaions occur a a consan rae per gamee and generaion, wih deleerious effec s in homozygoes and hs in heerozygoes, where h is he degree of dominance. We will 5

6 ignore episasis and linkage. Firs consider he siuaion where s and h are consan. As i is well-known, he expeced response o naural selecion for relaive finess is he corresponding finess addiive variance ( W s = V a, Fisher 930), and he rae of inbreeding depression is = [s( 2h)pq], () where he summaion is over all segregaing loci, q is he frequency of he deleerious allele and p = - q. For an infinie populaion, he per-generaion rae of finess decline from new muaion is W m =2 hs. Thus, since a he muaion selecion balance W m = W s, he equilibrium addiive variance for a hypoheical infinie populaion is V a = 2 hs. (2) The average number of copies conribued by a new no compleely recessive deleerious muaion before i is eliminaed is /hs, all hese copies occurring in heerozygosis, and can be denoed as is pervasiveness (Crow 979, García-Dorado e al 2003). Therefore, he overall frequency of deleerious alleles, adding up over loci, is q = /hs, and we ge he well known equilibrium equaion: [s( 2h)q] [s( 2 h)] / hs = (-2h)/h. (3) The equilibrium finess geneic properies in finie populaions For a finie populaion wih effecive size equal o he acual populaion number N, he heerozygous pervasiveness is approximaely he same as ha for he infinie one when selecion agains heerozygoes is he leading force (say hs>8n, Li and Nei 972). 6

7 For his reason, he approximaion q = /hs has ofen been used o infer he geneic properies of finess. However, as will be shown laer, his approximaion can induce an imporan bias depending upon he magniude of deleerious effecs and effecive populaion sizes. The overall frequency of deleerious alleles and of deleerious homozygous genoypes over loci: A he MSD balance under non-recurren muaion and assuming small average deleerious frequency per locus, each new muaion conribues on he average abou one new heerozygous genoype a he corresponding locus. Therefore, he per generaion increase of he expeced number of heerozygous loci per individual caused by new muaion is approximaely (2pq) mu =2. A equilibrium, he increase in (pq) due o new muaion will compensae is change from selecion ( (pq) sel ) and drif ( (pq) drifl ), so ha + (pq) drif + (pq) sel = 0, (4) as far as q remains small, so ha pq q, and (pq) drif = - (pq)/2n - q/2n, (5) (pq) sel q sel. (6) In finie populaions, he reducion in frequency due o selecion agains homozygous canno be ignored for small h values. Thus, a each segregaing locus, q sel = - qhs q 2 (-2h)s, (7) and, adding up over loci, q sel = - qhs q 2 (-2h)s, which can be wrien as 7

8 q sel = -s q {h K (-2h)}, (8) where K = q 2 / q (9) is he proporion of deleerious copies ha undergo selecion in he homozygous condiion, which, for single loci, is equal o he deleerious frequency q. In Appendix we obain he following approximae expression for K: K. (0) 4Nhs + 2 Ns + 2 Therefore, from equaions (4), (5) and (8), we obain q = 2N + hs + Ks( 2h), () which gives he expeced number of deleerious alleles per gamee, and q 2 = 2N K, (2) + hs + Ks( 2h) which represens he expeced number of loci for which any individual is homozygous for deleerious alleles. The equilibrium inbreeding depression rae for finess: Assuming, again pq q, and subsiuing () ino (), he inbreeding depression rae a he MSD balance for populaion size N can be approximaed as: 8

9 = 2N s( 2h) + hs + Ks( 2h), (3) The addiive and dominan componens of finess variance: For a populaion wih effecive size N, he per-generaion rae of finess decline a MSD equilibrium is due o. fixaion and amouns o D m = 2N su, where U is he probabiliy of final fixaion for a new muaion. The change in finess per generaion (-D m ) should equal he overall finess change due o reducions from inbreeding depression ( /2N) and new muaional inpu (2 hs) and o increase from response o naural selecion, which, from Fisher s heorem, is equal o he addiive variance for relaive finess (V a ). Thus: -2N su = Va - 2 hs - /2N, (4) so ha V a = 2 hs + -2N su (5A) 2N or V s( 2h) = 2 hs + (5B) + 2Nhs + 2NKs( 2h) a D m When N is no oo small N (i.e., for Ns>>), D m (i.e., 2N Us) is small and i can be ignored in he shor o medium erm, so ha V a 2 hs +. (6) 2N Adding up over loci, he dominance variance is V d = s 2 (-2h) 2 p 2 q 2 (Falconer and Mackay 996). Since, for q<<p, p 2 q 2 q 2, using equaion 2 we ge 9

10 V d K 2N 2 s (- 2h) + hs + Ks( 2h) 2 (7) Finess changes afer permanen reducion in populaion number Consider a populaion a MSD balance wih effecive number N, which a generaion 0 is suddenly reduced o a smaller sable number N. I will derive approximae predicions for he rae of change on mean and on addiive variance for finess as he populaion shifs o he new MSD balance ( shif approximaions ). The rae of inbreeding depression and he addiive variance during he shif: Firs I infer he finess geneic properies a boh he ancesral MSD balance (, V a and V d ) and he new one (, V a and V d ). Then, I assume ha he reducion in geneic diversiy occurring while he populaion adjuss o he new balance is due o alleles ha behave roughly as neural under he new smaller populaion number. For neural variaion, a fracion (-f ) of he individuals a generaion shows he geneic consiuion of he ancesral panmicic populaion a each locus. Therefore, we can inerpolae he inbreeding depression rae a a paricular generaion during he shif process from he ancesral and new equilibrium values using he corresponding inbreeding coefficien: = (-f ) + f (8) In he absence of new muaion, he evoluion of addiive variance under inbreeding could be prediced using he neural approximaion given by López-Fanjul e al. (2003) V a = V a (-f ) + cov ( 2, H) + 2 f (-f ) V d, 0

11 where cov ( 2, H) is he expeced covariance, added over loci, beween he squared average effec for gene subsiuion 2 (Falconer and Mackay 996) and he corresponding heerozygosiy H for differen hypoheical populaions derived wih size N from he ancesral populaion generaions ago. For q<<p, his can be approximaed as: where cov( 2,H) C f (-f ), C = 2 4 h( 2h) s, (9) + hs + Ks( 2h) 2N (see Appendix 2). In our case, he addiive variance undergoes an increase due o new muaion which, afer generaions, amouns o m, so ha V a = V a (-f ) + C f (-f ) + 2 f (-f ) V d + m (20) To express m in erms of he geneic properies of he wo equilibria, consider a fuure generaion a which he new equilibrium has been aained, wih V a, V d and C o be compued using N insead of N. Thus, he addiive variance afer addiional generaions should coninue being V a, so ha V a = V a (-f ) + C f (-f ) + 2 f (-f )V d + m. Therefore: m = V a f C f (-f ) - 2 f (-f )V d, and inroducing his expression ino equaion 20, we ge

12 V a = V a (-f ) + V a f + 2 f (-f ) (V d V d ) + f (-f ) (C - C ). (2) Finally, subsiuing equaion 5B ino his expression, we obain where D m and D m are he per generaion raes of finess decline due o deleerious fixaion a he ancesral and he new equilibrium, respecively, and is he ransiory excess in addiive variance above he inerpolaion beween V a and V a using f. This ransiory excess in addiive variance can be compued as f (-f )[2(V d - V d ) + (C - C )]. (23) When boh N and N are no oo small, D m and D m can be negleced, and V a can be approximaely compued as V a ' ' V a = 2 hs + (- f ) ( Dm ) + f ( Dm ) +, (22) 2N 2N' ' 2 hs + (- f ) + f +, 2N 2N' The per generaion rae of finess change: The change in mean finess from generaion o + while he populaion adjuss o he new balance can be compued considering he response o naural selecion, he deleerious muaional inpu and he inbreeding depression: W = V a - 2 hs - /2N. (24) Subsiuing equaions 8 and 22 ino 24, we ge: 2

13 W = (-f )[ (/2N /2N )-D m ] - f D m +. (25) Thus, measures he ransiory purging effec due o naural selecion on he corresponding excess of addiive variance. For N large enough ha /2N and D m can be ignored, and noing ha we obain (-f )(/2N ) = f + - f, W - (f -f - )- D m f +. Therefore, assuming ha per generaion changes are addiive, he mean finess a generaion is: W W 0 - f - i= D' m f i +. = (26A) In order o have selecion coefficiens ha remain consan hrough he process when expressed relaive o he average populaion finess, i may be more appropriae o consider he per-generaion changes in mean W as proporional raes of change, so ha W + =W ( - W ). In his siuaion, finess decays in a non linear way and, since per generaion changes are small, W + = W 0 W W 0 = W e, so ha W W 0 exp[- f - i= D' m f i + ]. = (26B) Thus, he finess decline is ha expeced from he ancesral inbreeding depression rae, plus he decline from fixaion of new deleerious muaions ha accumulaes as he 3

14 new equilibrium is aained, minus he purging effec of selecion on he ransiory excess of addiive variance induced by random drif of parially recessive deleerious alleles. Eqs. 26 have some heurisic value and are good approximaions even for small N, bu require compuing D m and, herefore, he deleerious fixaion rae from a diffusion approximaion which, for non addiive gene acion, mus be numerically calculaed. For N and N large enough ha boh D m and D m can be negleced, equaion 25 gives he following approach, which is compuaionally more convenien: W = (-f ) (/2N /2N ) +, and, assuming ha per generaion finess changes are addiive, W W 0 + i= [(-f ) (/2N /2N ) + ]. Or, for beween-generaion muliplicaive finess (see above), W + = W 0 exp [ i= { (-f ) (/2N /2N ) + }] (27) The change in mean under equal family conribuion Assume ha, during he shif process discussed in he previous secion, a managemen sraegy is esablished where each paren conribues wo offspring o he nex generaion (EC sraegy). Thus, he corresponding inbreeding coefficien (denoed f E ) should be compued using he effecive populaion size N e 2N. I will separaely consider he cases where naural selecion only occurs hrough differences in feriliy, or in viabiliy. 4

15 Feriliy: When naural selecion occurs only hrough differences in feriliy, equaing family conribuions (EC) implies eliminaing naural selecion. Then, he heerozygosiy increases by 2 due o new muaion and decreases by 2pq/4N (where he sum is over all loci) due o drif each generaion. This implies ha in he new balance 2pq=8N, and he corresponding inbreeding depression rae, obained from equaion, is =4N s(-2h), where, henceforh, he zero superscrip sands for he finess geneic properies of a populaion mainained by EC where naural selecion acs hrough feriliy differences. Using in equaion 8 gives = (- f E ) + f E. Furhermore, he feriliy decline from generaion o + is W = - 2 hs - /4N. Using hese wo expressions, he expeced average feriliy a any generaion under EC, when he per generaion finess decline is muliplicaive, is given by W = W 0 exp [ i= {- (- f E )/4N - f E /4N - 2 hs}], (28) which can be wrien as W = W 0 exp [- f E - f a E /4N -2 hs], 5

16 where f a E = i = E f is he accumulaed inbreeding coefficien wih EC. Once he new equilibrium is aained, he average finess will decline a a rae s due o coninuous fixaion of deleerious muaion, which represens he effecs of he inpu of new deleerious muaion (2 hs) plus ha due o inbreeding depression /4N = s(-2h). Viabiliy: If naural selecion acs exclusively hrough viabiliy differences, EC implies ha selecion acs only on he addiive variance expressed wihin sib-families, which is half he populaion addiive variance. Therefore, he efficiency of naural selecion corresponds o ha expeced in a populaion wih half he acual addiive variance, i.e. wih selecion coefficiens s*=s/ 2. Henceforh, he aserisk superscrip sands for he finess geneic properies of a populaions mainained by EC where naural selecion acs hrough viabiliy differences. Using s* insead of s in equaions 0, and 2, and subsiuing ino equaion, we obain ha he inbreeding depression rae expeced a he new balance is * = s( 2h), + hs * + K * s *( 2h) 4N' where K* =. 8N' hs * + 4 N ' s * + 2 Noe ha he probabiliy of deleerious fixaion decreases wih increasing values of he compound parameer N e s. Since, in he new equilibrium, N e s* =N s 2, he long 6

17 erm rae of viabiliy decline due o deleerious fixaion under EC (i.e., - D ' m *) will be smaller han under no managemen (NM). Since he per generaion rae of finess change a he new equilibrium is - D ' m * = V a */2-2 hs - */4N, he corresponding addiive variance should be V a * = 4 hs + */2N - 2 D ' m *. Thus, from equaion 2 we ge V a * = (- f E ) ( 2 hs + Dm ) + f E (4 hs + /2N 2 D ' m *) + */2, (29) 2N where */2 represens he ransiory purging under EC, where and * = f E (- f E )[2(V d - V d *) + (C - C *)], V d * 2 2 s ( 2h) K *, + hs * + K * s *( 2h) 4N' Since C* = 2 4 h( 2h) s. + hs * + K * s *( 2h) 4N' * = (- f E ) + f E *, and considering ha he rae of decline from generaion o + wih EC is given by 7

18 W* = V a */2-2 hs - */4N, we obain W* = (- f E ) ( Dm ) 4N 4N' 2 hs + f E ( '* 4N' '* D ' m *) + */2 4N' and, rearranging, W* = (- f E ) ( Dm ) 4N 4N' 2 hs E - f D ' m * + */2. (30) reduces o For an ancesral populaion large enough ha D m and /4N can be ignored, his W* -(- f E )( 4N' E + hs ) - f D ' m * + */2. Therefore, noing ha wih EC (-f E )(/4N ) = f + E - f E, adding up over generaions, he average finess a generaion becomes W * = W 0 exp[- f E -(- i= f E ) hs - D ' m * i= f E + = */ 2 ], or, using he accumulaed inbreeding ( f a E ), W * = W 0 exp[- f E -(- f E a ) hs - D ' m * f E a + = */ 2 ]. Similarly, using equaion 26B for he average viabiliy under NM, we obain ha EC gives larger average viabiliy a generaion han NM when 8

19 f + D m f a - > f E + D ' m * f E a + (- f E a ) hs - */ 2. (3A) = = For N large enough ha he equilibrium rae of deleerious fixaion can be ignored, he condiion for EC giving larger viabiliy han NM reduces o (f - f E ) > (- f E a ) hs + { = - = */ 2 }, (3B) The lef side, (f - f E ), has a maximum for some inermediae f value, and hen decreases o zero. On he conrary, (- f a E ) increases wih up o a limi, which is larger and akes longer o be aained for larger N values. Therefore, seing aside he effec of purging, his managemen sraegy should be advanageous for he early generaions during he shif. However, unless h values are very small, EC will become disadvanageous as f E approaches one. This disadvanage requires a relaively large N value o be imporan, and will ake a long ime o emerge. Equaions 3A-3B also imply ha his disadvanage will increase wih increasing h values, as hese will produce small and large hs. Noe, however, ha large hs implies large V a * (equaion 29), i.e., since he long-erm disadvanage is due o a greaer load from segregaing deleerious muaions, i can be poenially recovered due o naural selecion afer NM is resored. As shown above, he long-erm deleerious fixaion rae (equaion 3A), is smaller under EC han under NM. The purging erm { = - = */ 2 } goes o zero as increases, bu i may be posiive firs and negaive laer, implying ha delayed purging under EC can conribue o a middle-erm advanage for his managemen sraegy. 9

20 For N no oo small, when boh Dm and Dm * can be negleced, equaion 30 reduces o he compuaionally more convenien expression W* = (- f E ) ( ) 4N 4N' hs + */2, and W + = W 0 exp [ i= { (- f E ) ( ) 4N 4N' hs + */2}]. (32) PEDICTIONS AND DISCUSSION Figure Checking he analyical predicions for he Muaion-Selecion-Drif Balance Figure shows predicions for he inbreeding depression rae given by equaion 3 as a funcion of h, where s = 0.02 and differen panels are for differen populaion numbers. A predicion based jus on he pervasiveness of he allele compued from is heerozygous effec (i.e., assuming K=0) is given for comparison. I shows ha he consrain on posed by selecion agains homozygous becomes highly relevan for h<0. in large populaions. Predicions compued from diffusion heory are also given in Figure and show ha our approximaion is excellen in mos condiions. As an excepion, our approach produces downwardly biased predicions for N = 00, suggesing ha i overesimaes he efficiency of naural selecion when Ns = 2. This can be due o he inadequacy, in his case, of he assumpion q<<p when accouning for naural selecion (equaion 6). The predicions illusrae ha he equilibrium inbreeding depression rae increases almos linearly wih decreasing h values for small populaions. For large populaions, 20

21 however, becomes appreciable only for small h values, and, hen, selecion agains homozygous becomes a relevan limiing facor. Figure 2 Figure 2 (upper panel) gives predicions for he inbreeding depression rae obained using eiher our approximaion or he diffusion approach. Boh are averaged over plausible disribuions for muaional effecs as in García-Dorado There has been exensive discussion of he rae of deleerious muaion and he corresponding effec disribuion (García-Dorado e al. 2004). Basically, experimens by Mukai and co-workers wih Drosophila melanogase implied almos one deleerious muaion per new zygoe, wih average homozygous effec around a few percen, while laer reanalysis and newer experimens gahered evidence ha he rae of muaion wih deleerious effec relevan in he conex of his analysis is usually an order of magniude below he one proposed by Mukai, wih larger average homozygous effecs on he order of 0 -. Here I consider hree muaional parameer ses, derived from Drosophila daa, for which a more explici jusificaion can be found in García-Dorado One is a sensiive case (S c ) corresponding o he Mukai view, where finess is muaionally sensiive, i.e., where mos muaions have a relevan deleerious effec, alhough his is usually iny or mild (he precise raes and effec disribuion I use are hose of model (c) in Fernandez and Caballero 200, in order o allow appropriae comparison o heir simulaion resuls). The oher wo are oleran cases, where he deleerious effec of mos muaions, alhough migh be large enough as o consrain fixaion probabiliies in he evoluionary ime-scale, are oo small o make a relevan conribuion o he geneic properies sudied here. In boh oleran cases, a small fracion of muaions have mild o severe deleerious effec. One is T model in García- Dorado 2003, and he oher one (T h ) is case (h) in Fernández and Caballero 2003, which is similar o T wih lower kurosis for s. 2

22 Our approximaions were averaged over 0 4 muaions sampled from he corresponding join disribuions for s and h (see figure legend for muaional parameers). In general, hey are in good agreemen wih he diffusion ones. This implies ha muaions wih small s and h values, for which our mehod gives downwardly biased esimaes, do no make a relevan conribuion o he overall inbreeding depression rae, so ha he approximaion inegraed over he disribuion of muaional effecs is highly reliable compared o he corresponding diffusion approximaion (figure 2, upper panel). The figure illusraes a increase ha is roughly linear on he decimal logarihm of he effecive populaion number. Figure 2 also gives predicions for addiive (middle panel) and dominance (lower panel) variances obained from equaions 6 and 7, boh averaged for he disribuions corresponding o he differen muaional parameer considered, ogeher wih he corresponding diffusion predicions (see above and figure legend). Addiive variance approximaions are excellen, excep for small populaions wih S c muaional parameers. This bias is due o he use of he approximaion in equaion 6, where he rae of finess decline from deleerious fixaion in he new equilibrium (which is relevan in his paricular case, see García-Dorado 2003) has been negleced. The reason is ha, in his siuaion, muaions wih very small effec, which can become fixed by drif in small populaions, occur a a high rae. This bias could be correced by using equaion 5. The predicions for he dominance variance are reliable for large populaions, bu are over-esimaes for low o moderae size populaions, he bias being imporan for he S c case wih N on he order of he hundreds. This should again be ascribed o he inadequacy of he assumpion q<<p (equaion 6) in his siuaion, where drif is relaively large. 22

23 Figures 3, 4 Predicions for he addiive variance and he mean for finess during he shif process under No Managemen. Figure 3 gives he prediced addiive variance compued using his approximaion for he S c model (see parameer in Figure 2) for a 0N generaion period afer size reducion. Four siuaions, combining N=0 3 or 0 5 and N =0 or 50, are considered. Figure 4 is equivalen for T muaional parameer. In all hese cases, he addiive variance reached a maximum value afer abou 2N generaions (f 0.64), and hen decreased o approach a new balance wih larger addiive finess variance. Under boh models, he prediced increase in addiive variance was higher for he larger ancesral populaion size (N=0 5 ) and for he more drasic size reducion (N =0). Figure 5 Figure 5 gives differen predicions for average finess hrough he firs 50 shif generaions under no managemen. Average finess prediced using equaion 27 is considerably above he neural expecaion derived from he iniial inbreeding depression rae (W = W 0 exp [-f ]) and much higher han ha including he muaional inpu of deleerious effecs (W = W 0 exp [-f - i= 2 hs ]). The difference beween hese predicions is larger for S c han for T muaional parameers, as he former gives larger and larger muaional deleerious inpu. Wang e al. (999) obained sochasic simulaion resuls for N =50 saring from a populaion a a recurren muaion-selecion balance wih S c -like muaional parameer ( =0.5, E(s)=0.05, E(h)=0.36, s exponenially disribued, h=exp[-ks]). I have obained equaion 27 predicions for his case using ancesral geneic properies compued for he recurren muaion model ( and compued from equaions, 9 and 23, where q values were compued from equaion in Crow & Kimura 970 for 5800 loci as in 23

24 Wang e al. 999). Our predicions are in good agreemen wih simulaion resuls in heir Table 2 up o generaion 50 (differences below 2%). From generaion 50 o 00, however, he rae of viabiliy decline in Wang e al. doubled, which is unexpeced, paricularly under he assumed muliplicaive finess model. On he conrary, our predicions indicae ha, by generaion 00, viabiliy had recovered half of he previous decline. This difference can be parly ascribed o linkage in Wang e al. simulaions, where he whole genome lengh is 2 M. As Wang e al. poined ou, associaive overdominance induced by selecion in he presence of linkage for h<0.5 may reduce he effec of purging. Noe ha in he overdominan equilibrium he addiive geneic variance is zero, so ha associaive overdominance, alhough i may induce considerable segregaing load, should be expeced o reduce addiive variance in he long-erm and, herefore, o diminish he efficiency of purging. However he above difference should also be parly a consequence of he use of equaion 27 (which neglecs he finess decay from deleerious fixaion) insead of he analyically less racable 26B expression. This may become relevan in he long-erm (i.e., for high i= f i in equaion 26B), paricularly for S c -like muaional parameer, for which D m = for N =50 (compued from diffusion heory). Therefore, for small N values and S-like muaional parameers, (say N abou 0 and sensiive cases, see García-Dorado 2003), he rae of finess decline from deleerious fixaion becomes relevan, and using equaion 27 (insead of equaion 25, or is approximaed expression in equaion 26B) may underesimae he long-erm finess decline. Equal Conribuion (EC) vs. No Managemen (NM) during he shif process under differen muaional parameer 24

25 We use Eqs. 27, 28 and 32 o predic he change in mean feriliy or mean viabiliy under NM or EC for differen populaion numbers. In each case, predicions are averaged over 0 4 (s,h) values randomly sampled from he corresponding disribuion, according o he muaional parameer (see legend in Figure 2). Figure 6 Figure 6 shows predicions for he cases simulaed by Fernández and Caballero (200) and given in heir figure (a,b,c panels). These refer o an ancesral populaion where 5800 loci segregae wih he frequency expeced a he balance beween naural selecion and recurren muaion, as in Wang e al. 999, bu linkage was no considered in his case. S c muaional parameer are used and he figure gives he evoluion of mean finess when he effecive populaion size is reduced o a sable value N =25. I obain predicions comparable o hese simulaion resuls following wo differen approaches. In he upper panels of Figure 6, I have used our predicions for non recurren muaion assuming N=000. This N value was chosen because i provides an ancesral inbreeding depresión rae ( =.49) very close o he value for non recurren muaion in Fernández and Caballero work ( =.67). Furhermore, i also gives an equilibrium number of segregaing loci (7383) ha is of he order of ha used in he simulaion procedure. In he middle panel, I have obained predicions using ancesral geneic properies compued for he recurren muaion model, wih and compued from equaions and 23, and using V d = s 2 (-2h) 2 [q(-q)] 2 and C = 4h(-2h) s 2 q wih q values obained from equaion in Crow & Kimura ( 970) for 5800 loci, as for he Wang e al. 999 case discussed above. The lower panel give Fernández and Caballero simulaion resuls. Panel on he lef compare resuls under NM and EC for naural selecion acing on feriliy, and righ side panel do he same for viabiliy. The presen shif approximaions give quie precise finess predicions for he EC-feriliy case, where here is no selecion, bu NM and EC-viabiliy predicions are downwardly 25

26 biased by up o 0%. However, he good qualiaive agreemen beween simulaion and predicions in boh cases is remarkable, indicaing ha, despie he approximaions involved, our equaions have considerable predicive power. A qualiaive agreemen was also found wih ransiion marix predicions by Schoen e al. (998) and by Theodorou and Couve (2003), alhough he absence of some deails regarding he iniial ancesral populaion and he disribuion of h prevens formal comparison. Figure 7 Figure 7 gives he evoluion of finess under S c muaional parameer for NM and for feriliy or viabiliy under EC. Each panel represens a differen siuaion regarding N and N values. For feriliy, EC is a an advanage in he shor erm for large N and small N values (cases wih N/N 00), as his promoes large ancesral inbreeding depression raes, and high inbreeding during he shif. Regarding viabiliy, EC is always a an advanage for a considerable period if he ancesral populaion was large, bu boh he lengh of ha period and he magniude of he advanage become insignifican for smaller ancesral populaions, unless he new size is very small. Figure 8 Figure 8 gives analogous resuls for T muaional parameers. In his case, where he rae of relevan deleerious muaion is low and he average deleerious effec is moderae and predominanly recessive, EC is a an advanage for feriliy in he shorerm for a larger specrum of condiions (cases wih N/N 40), because he addiive variance is smaller and, herefore, purging is less efficien. Regarding viabiliy, EC was always a some advanage during he period considered, alhough his only was imporan when he ancesral populaion was large. Drosophila experimenal sudies have been performed in order o assess he advanage for EC sraegies. Borlase e al. (993) repored ha, afer generaions wih N =8, heir populaion had relaive finess 0.23 if mainained wih NM and 0.26 if EC was pracised. This is qualiaively consisen wih our Shif approximaions, 26

27 which predics in his case advanages abou for relaive feriliy and relaive viabiliy for all muaional parameer values considered above. Thus, using S c muaional parameer wih N=0 6 and N =8, he approximaion predics relaive finess 0.4 afer generaions wih NM, while wih EC i predics 0.28 when selecion acs hrough viabiliy or 0.25 when i occurs hrough feriliy. T models predic similar differences, bu wih higher finess values (abou 0.5 wih EC). However, i is well known ha inbreeding depression raes have considerable beween-populaion variabiliy, and recen resuls sugges ha he rae of severely deleerious muaion wih small h could be underesimaed in muaion accumulaion experimens due o purging (García-Dorado e al., 2006). Thus, average deleerious effecs could be slighly above hose corresponding o T parameers, leading o subsanially higher inbreeding depression raes a equilibrium. As an ad hoc approximaion, a T-like muaional parameer se wih =0.03, s exponenially disribued wih average 0.3, and he same relaionship beween h and s as in T model, predics for he Borlase e al. case (N=0 6, N =8, =) relaive finess 0.26 under EC (boh for viabiliy and feriliy) and 0. under NM. Thus, Borlase e al. resuls are consisen wih our predicions under a variey of muaional parameers, boh for feriliy and viabiliy. Rodríguez-Ramilo e al. (2006) mainained Drosophila populaions wih N =00 or 20 for 35 generaions using NM or EC. In large populaions, relaive viabiliy declined o abou 0.8, wih no deecable difference beween mehods, while wih N =20, relaive viabiliy declined o abou 0.6, showing a small bu consisen advanage for EC (see heir figure 5B). These resuls are qualiaively consisen wih our approach assuming N=0 6 and T muaional parameers, which predics relaive viabiliy 0.90 under EC and 0.84 under NM for N =00, and 0.69 under EC and 0.57 under NM for N =20. The S c muaional parameers predic a similar advanage paern, bu wih oo 27

28 small average viabiliy (abou 0.55 for large populaions and down o 0. for small ones). Therefore, inspecion of Figures 7 and 8, as well as empirical resuls, show ha EC is a an advanage regarding finess for a wide range of siuaions, paricularly if managemen is expeced o be necessary for less han abou 20 generaions. However, analyical resuls and pracical reasons make i convenien o separaely consider feriliy and viabiliy. Thus, for relaively low N / N raios, EC may convey subsanial disadvanage for feriliy if derimenal muaions occur a a high rae, which is due o relaxed selecion on feriliy (lef panels in Figure 7 for S c muaional parameers). On he oher hand, feriliy rais migh be prone o undergo adapaion o capive condiions, hrough modified breeding behaviour, increased opimum lier sizes ec, so ha relaxed selecion on feriliy migh become an addiional advanage for EC managemen. In any case, i should be remembered ha our conclusions jus refer o direc effecs on finess, and ha an ancesral balance beween deleerious muaion and drif is assumed and no forces oher han hese are considered during he process. Several mechanisms can lead o larger ancesral geneic variance and/or inbreeding depression rae for finess, hus affecing he validiy of he above predicions. Furhermore, for populaions saring wih high reproducive poenial and good viabiliy, oher consideraions, such as he improved mainenance of adapive poenial and variabiliy, or he slower rae of adapaion o capive condiions, may be more relevan when deciding on he advanages of using EC managemen. 28

29 ACKNOWLEDGEMENES I am graeful o Armando Caballero and Carlos López-Fanjul for helpful discussion and commens on he manuscrip. This work was suppored by grans CGL /BOS and INIA (CPE C2) from Miniserio de Ciencia y Tecnología. 29

30 LITERATURE CITED Borlase, S.C., D.A. Loebel, R. Frankham, R.H. Nurhen, D.A. Briscoe and G.D. Daggard. e al., 993 Modeling problems in conservaion geneics using capive Drosophila populaions: Consequences of equalizaion of family sizes. Conservaion Biology 7:22-3. Crow, J.F., 979 Minor viabiliy muans in Drosophila. Geneics 92: s65-s72. Crow, J.F. and M. Kimura, 970 An Inroducion o Populaion Geneics Theory. New York: Harper and Row. Falconer DS, Mackay T, 996 Inroducion o Quaniaive Geneics. 4 h ed. Longman Inc, Essex, England. Fernández, J. and A. Caballero, 200 Accumulaion of deleerious muaions and equalizaion of parenal conribuions in he conservaion of geneic resources. Herediy 86: Fisher, R.A., 930 The Geneical Theory of Naural Selecion. Clarendon Press, Oxford. Frankham, R., J.D. Ballou and D.A. Briscoe, Inroducion o Conservaion Geneics. Cambridge Universiy Press. Cambridge UK. García-Dorado, A., 2003 Toleran versus sensiive genomes: he impac of deleerious muaion on finess and conservaion. Conservaion Geneics 4: García-Dorado, A., C. López-Fanjul, and A. Caballero, 2004 Raes and effecs of deleerious muaions and heir evoluionary consequences, pp in Evoluion: From Molecules o Ecosysems, edied by A. Moya and E. Fon. Oxford Universiy Press, Oxford. García-Dorado, A., V. Ávila,3, E. Sánchez, A. Manrique and C. López-Fanjul,4, 2006.The build up of muaion-selecion balance in populaions of moderae size: experimenal resuls wih Drosophila. (In revision for Evoluion) 30

31 Kimura, M., 969 The number of heerozygous nucleoide sies mainained in a finie populaion due o seady flux of muaions. Geneics, 6, Li, W. H. and M. Nei, 972 Toal number of individuals affeced by a single deleerious muaion in a finie populaion. Am. J. Hum. Gene. 24: López-Fanjul, C., A. Fernández and M.A. Toro, 2003 The effec of neural nonaddiive gene acion on he quaniaive index of populaion divergence. Geneics 64: Lynch M., J. Conery and R. Bürger, 995 Muaional meldown in sexual populaion. Evoluion 49: Rodriguez-Ramilo, S.T., P.Moran and A. Caballero, 2006 Relaxaion of selecion wih equalizaion of parenal conribuions in conservaion programs:an experimenal es wih Drosophila melanogaser. Geneics 72: Schoen, D.J., J.L. Davis and Thomas M. Baaillon, 998 Deleerious muaion accumulaion and he regeneraion of geneic resources. Proc. Nal. Acad. Sci. 95: Theodorou, K. and D. Couve, 2003 Familial versus mass selecion in small populaions. Gene. Sel. Evol. 35: Wang, J., W.G. Hill, D. Charlesworh and B. Charlesworh, 999. Dynamics of inbreeding depression due o deleerious muaions in small populaions: muaions parameers and inbreeding raes. Gen Res. Camb. 74: Wrigh, S., 938 Size of populaion and breeding srucure in relaion o evoluion. Science 87:

32 APPENDIX Assuming Ns>5, Li & Nei (972) provide approximaions for he whole expeced number of heerozygous (n ) and homozygous (n 2 ) individuals ha will carry a new deleerious muaion while i will segregae in he populaion under wo differen assumpions (h>0.3, h=0, see below). Therefore, he raio q 2 / q (where summaion is over generaions) can be compued as q 2 q = n n + n 2 / 2, This also gives he raio of he expeced q 2 value o he expeced q value a single generaions. Thus, we compue K = n n + n 2 / 2. a) For h > 0.3, naural selecion acs mainly hrough he muan effec in he heerozygous sae, so ha n = /hs and n 2 = /(8Nh 2 s 2 ) (equaions 6b, 8b in Li & Nei 972). Therefore, denoing by K hs he K value when hs>0.3, 2 2 /(8Nh s ) K hs = =, 2 2 /(8Nh s ) + /(2hs) 4Nhs + b) For h=0, naural selecion only limis q values hrough selecion agains homozygous, and Li and Nei (972, equaions 7b y 9c) give n = 2N /( 2Ns), n 2 =/(2s), 32

33 and, denoing by K s he K value when h = 0, K s = /(2s) = /(2s) + N /(2Ns) + 2 Ns. We will use K K hs + K s = 4Nhs + 2 Ns + 2 as a general approximaion, where K is mainly limied by he more relevan deleerious effec (s or sh) in each case (i.e., he driving facor consraining q 2 / q is he homozygous deleerious effec when K s << K hs and he heerozygous deleerious effec when K hs << K s ). This K approximaion has been compared o q 2 / q values compued from diffusion approximaion, showing a good behaviour for quie general s, h, N values (resuls no shown). We noe ha when s goes o zero, K goes o, as i is expeced from diffusion heory. APPENDIX 2. The expeced covariance cov ( 2, H) a any segregaing locus beween he squared average effec of a gene subsiuion ( = s/2 + (2q-)(-2h)s/2, Falconer and Mackay 996) and he corresponding heerozygosis in differen populaions randomly derived and mainained wih effecive size N from he ancesral populaion generaions ago can be wrien as cov ( 2,H) = [(s/2 + (2q-)(-2h)s/2) 2, 2pq]. This gives cov ( 2,H) = cov (s 2 /4, 2pq) + cov [s(2q-)(-2h)s/2), 2pq] + cov [((2q-)(-2h)s/2) 2, 2pq]. 33

34 Noe ha cov (s 2 /4, 2pq)=0, and, for q<<p, cov [(2q-) 2, 2pq] cov [-4q, 2pq]. Furhermore, for q<<p, cov (q, 2pq) 2V (q), where V sands for variance. However, he las approximaion ignores ha, alhough for any q 0 ancesral q value, by generaion we have E(q) = q 0, bu E(2pq) = 2p 0 q 0 (-f ). To correc for his approximaion we use Cov (q, 2pq) 2(-f )V (q). As i is well known, V (q) = pqf, so ha, ignoring again quadraic q erms, pq q and Cov(q, 2pq) 2(-f ) f q. Therefore, Cov( 2,H) 2(-2h)s 2 f (-f ) q - 8((-2h)s/2) 2 f (-f ) q And, rearranging, Cov( 2,H) 4h(-2h)s 2 f (-f ) q, which, adding up over loci, gives Cov( 2,H) 4h(-2h)s 2 f (-f ) q Or cov( 2,H) C f (-f ) Where C = 4h(-2h)s 2 + hs + Ks( 2h) 2N 34

35 Figure legendes: Figure.- Equilibrium raes of inbreeding depression agains he coefficien of dominance h. Diffusion predicions (coninuous line), equaion 3 predicions (broken line wih filled circles) and equaion 3 predicion assuming K=0 (broken line wih empy circles). In all cases s = 0.02 and =. Panels are for differen effecive populaion numbers (a: N=0; b: N=00; c: N=0 3 ; d: N=0 4 ) Figure 2.- Equilibrium geneic parameers for differen muaional properies agains. log 0 (N). Diffusion (coninuous line) and approximaed (broken line) predicions for he rae of inbreeding depression (, upper panel), addiive variance (V a, middle panel) and dominan variance (V d, lower panel) agains Log 0 (N) for hree differen ses of muaional parameers: a) Sensiive S c case, as (c) case in Fernández e al 200: =0.5, s exponenially disribued wih average effec E(s) = 0.05, h uniformly disribued beween 0 and exp[- ks] wih average E(h)=0.35. Filled circles. b) Toleran T h case, as (h) case in Fernández and Caballero 200: =0.03, s gamma disribued wih shape parameer 2.3 and average effec E(s) = 0.264, h uniformly disribued beween 0 and exp[-ks] wih average E(h)=0.20; Empy squares. c) Toleran T case, as in García-Dorado 2003: =0.03, s exponenially disribued wih average effec E(s) = 0.224, h uniformly disribued beween 0 and exp[-ks] wih average E(h)=0.20. Empy riangles. Figure 3.- Addiive variance wih NM (no managemen) agains generaion number during 0N shif generaions for S c muaional parameers. 35

36 Figure 4.- Addiive variance for NM (no managemen) agains generaion number during 0N shif generaions for T muaional parameers. Figure 5.- Differen predicions for he finess average agains generaion number during he shif o a new balance for N =25 under no managemen. Thick line: shif approximaions ; Doed line: predicions derived from he ancesral inbreeding depresión rae. Thin line: predicions derived using and he muaional deleerious inpu. Figure 6. -Finess average agains generaion number for he case simulaed by Fernández and Caballero (200): (50 shif generaions for N = 25) predicions and simulaion resuls. Solid lines: NM (no managemen); Doed lines: EC (equal conribuion). Simulaion resuls from Fernández and Caballero 200. Figure 7. -Predicions for average finess agains generaion number during shif from N o N' for S c muaional parameers. Thick line: NM (no managemen); Doed line: EC (equal conribuion) when naural selecion acs hrough feriliy; Thin line: EC (equal conribuion) when selecion acs hrough viabiliy. Figura 8.- Predicions for average finess agains generaion number during shif from N o N' for T muaional parameer. Thick line: NM (no managemen); Doed line: EC (equal conribuion) when selecion acs hrough feriliy; Thin line: EC (equal conribuion) when selecion acs hrough viabiliy. 36

37 Figure a : N = 0 b : N = δ δ h h.6 c : N = 0 3 d : N = δ δ h h.6

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