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Recessive Mutations and the Maintenance of Sex in Structured Populations


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Copyright2001 by the Genetics Society of America

Recessive Mutations and the Maintenance of Sex in Structured Populations

Aneil F. Agrawal* and J. R. Chasnov

*Department of Biology, Indiana University, Bloomington, Indiana 47405 andDepartment of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong

Manuscript received January 19, 2001 Accepted for publication March 26, 2001


The evolutionary maintenance of sexual reproduction remains a controversial problem. It was recently shown that recessive deleterious mutations create differences in the mutation load of sexualvs.asexual populations. Here we show that low levels of population structure or inbreeding can greatly enhance the importance of recessive deleterious mutations in the context of sexual vs. asexual populations. With population structure, the cost of sex can be substantially reduced or even eliminated for realistic levels of dominance.


LL else being equal, asexual populations have a two- gous state where they can be more efficiently eliminated fold fitness advantage over their sexual counter- by selection. Population structure or inbreeding in-parts (Maynard Smith 1971, 1978; Williams 1975; creases the proportion of homozygotes and thus allows Bell1982). Among a variety of possible explanations selection to be even more efficient in sexual populations for the continued prevalence of sexual reproduction (Crow1970). We show that biologically reasonable lev-(Kondrashov 1993; Hurst and Peck 1996; Barton els of population subdivision greatly expand the param-andCharlesworth1998), the mutational determinis- eter space over which sexuals experience reduced muta-tic hypothesis is one of the leading proposals. If the tion load relative to asexuals.

mutation rate is sufficiently high and deleterious muta- Consider a single locus that mutates from the wild-tions interact synergistically, then sexual populawild-tions type allele,A, to the deleterious allele,a, with probability can clear mutations more efficiently and thus enjoy a u.The fitnesses of the three genotypesAA,Aa, andaa

much higher mean fitness at mutation-selection balance are 1, 1⫺hs, and 1⫺s, respectively. The degree of domi-than asexual populations (Kimura and Maruyama nance of thea allele is described byh, where values of 1966;Kondrashov1982, 1988;Charlesworth1990). 0 and 1 represent completely recessive and dominant Although there is some theoretical support for syner- alleles, respectively. The frequencies ofAandaalleles in gistic epistasis (Szathmary 1993; Peck and Waxman the sexual population are given bypandq, respectively. 2000), there is little experimental support for this type Using Wright’s inbreeding coefficient,f, as a measure of gene interaction (Willis 1993; Elena and Lenski of population subdivision, the genotype frequencies be-1997). In contrast, there is wide support for the domi- fore and after selection are given in Table 1. The calcula-nance of wild-type alleles over their deleterious counter- tions implicitly assume hard selection such that a deme parts (Lynchand Walsh1998). Chasnov (2000) re- contributes to the metapopulation in proportion to its cently showed that differences in mutation load mean fitness.

(Haldane 1937; Mu¨ ller 1950; Crow 1970) due to Allowing for mutation to occur between generations, recessive deleterious mutations alone (i.e., no epistasis) the frequency of the mutant allele in the next genera-can result in a substantial advantage to sexual reproduc- tion is

tion over asexual reproduction under some conditions.

This result is perhaps unsurprising as dominance creates q⬘ ⫽u⫹[1⫺ush(1u)sf(1⫺h(1u))]qs(1f)(1⫺h(1u))q




mutational synergy within loci (whereas epistasis creates (1)

it among loci). The advantage to sex exists because

heterozygous parents can produce homozygous off- where the average fitness of individuals in the sexual spring through sexual reproduction but not through population with respect to this locus is

asexual reproduction (except by mutation). Relative to

wsex⫽1⫺s[2hf(1⫺2h)]qs(1⫺f)(1⫺2h)q2. (2)

asexual populations, a greater fraction of segregating mutations in sexual populations exist in the

homozy-The equilibrium frequency of theaallele at mutation-selection balance can be found by solving (1) forq⬘ ⫽q

(Crow 1970;Crow and Kimura 1970). One solution Corresponding author:Aneil F. Agrawal, 1001 E. 3rd St., Department

isq⫽ 1, and the other two solutions are roots of the of Biology, Indiana University, Bloomington, IN 47405-3700.

E-mail: aagrawal@bio.indiana.edu quadratic equation



Single-locus genetic model

Genotype AA Aa aa

Fitness 1 1⫺sh 1⫺s

Frequency in zygote p2pq f 2pq(1f) q2pq f

Frequency after selection p

2pq f wsex

2pq(1⫺f)(1⫺sh) wsex

(q2pq f)(1s) wsex

Note thatpq⫽1 andwsex⫽1⫺s(2hpqq2)⫺s(1⫺2h)pq f.


f)(1⫹u))qu⫽0 (3) where n is the number of diploid genes per genome and具l典is the average contribution to load from a single (Crow1970). An approximate solution to (3) foruⰆ1

locus. Forsu(2⫺u) andhu/s, the mutation load and h, s, fu is determined by neglecting terms of

per locus for asexual reproduction is 2u(Kimuraand orderu2and assumingqO(u):


Taking the same value of u, s, and h at each locus


s[fh(1⫺f)]. (4) and definingUnuas the haploid genome-wide muta-tion rate, the relative fitness of sexual individuals to The mutation load per locus,lsex⫽1⫺wsex, correspond- asexual individuals is

ing to this solution is


f U

fh(1⫺ f)

. (7)


f⫹ 2h(1⫺ f)

fh(1⫺ f). (5)

Comparison of this analytical approximation to exact To determine when mutation load provides an

advan-numerical evaluations demonstrates reasonable agree-tage to sexual reproduction, we consider the ratioWsex/ ment provided f,h, ands are not too small (Figures 1

Wasex, where Wsex and Wasex are the genome-wide mean and 2). How is (7) to be interpreted? First, (7) shows no

zygote fitnesses (with respect to load) for sexual and

advantage to sex whenf⫽0 (random mating) because of asexuals, respectively. Any value ofWsex/Wasex⬎ 1

indi-the inherent assumptionhu/s.The casef⫽0 is solved cates that sexual populations have an advantage over

in detail byChasnov(2000). Second, to leading order asexual populations with respect to mutation load.

inu, the advantage to sex is independent of the selection WhenWsex/Wasex⬎2, this advantage completely

compen-coefficient,s. Third, the advantage to sex ranges from sates for the twofold cost of sex.

a maximum when hf of approximately exp(U) to a Assuming loci are in linkage equilibrium and no

epis-minimum whenhfof exp(fU/h). tasis,Chasnov(2000) has shown that

The value ofWsex/Wasex depends strongly on the

ge-nome-wide mutation rateUnu.If this value is on the

Wsex/Wasex≈exp[n(具l典asex⫺ 具l典sex)], (6)

Figure 1.—Mutation rate

and the advantage of sexual re-production. The mean fitness of sexual relative to asexual in-dividuals,Wsex/Wasex, is plotted

as a function of the genome-wide mutation rate,Unu.Six different levels of population structure, f, are plotted. n ⫽ 105,s0.01, andh0.1 while uwas varied from 1⫻10⫺6to

1.5⫻10⫺5. Symbols show exact

values of Wsex/Wasex found by


Figure2.—Population struc-ture and the advantage of sex-ual reproduction. The mean fitness of sexual females rela-tive to asexual individuals, Wsex/Wasex, is plotted as a

func-tion of the extent of popula-tion subdivision or inbreeding, f. n⫽105,u10⫺5, andUnu⫽ 1. (a) s⫽ 0.01 and six levels of the dominance coef-ficient,h, are shown. (b)h⫽ 0.1 and three levels of the selec-tion coefficient,s, are shown. Dashed lines connect exact val-ues ofWsex/Wasexfound by

nu-merical evaluation. Solid lines represent the analytical ap-proximation shown in (7).

order of 0.1 or smaller, then Wsex/Wasex ≈ 1. As U in- there is little advantage to sex, and Wsex/Wasex ⫽ 1.07

creases,Wsex/Wasexincreases exponentially when there is without population structure (f⫽0). With some

popula-some degree of population structure (Figure 1). For tion structure (f ⫽0.1), the sexual population experi-values of U on the order of 1, Wsex/Wasex can take on ences a much reduced mutation load,Wsex/Wasex⫽1.71.

values substantially greater than unity. As shown by In general,Wsex/Wasexis larger for smaller values of the

Chasnov(2000), when there is no population structure selection coefficient,s, although this effect diminishes (i.e.,f⫽0),Wsex/Wasexis highly sensitive to the degree of with increasing levels of population subdivision

(Fig-dominance,h(Figure 2). Without population structure, ure 2b).

values ofWsex/Wasex Ⰷ 1 occur only with extremely low What are the actual values of the relevant parameters?

There is general agreement that the mutation rate for dominance [i.e.,hO(

u/s); seeChasnov2000].

prokaryotes such asEscherichia coliis much less than unity Small increases in the amount of population

struc-whereas it is greater than unity for long-lived eukaryotes ture,f, can cause substantial increases inWsex/Wasex.

Fig-such as humans (Keightleyand Eyre-Walker1999; ure 2a shows that the addition of population structure

Lynchet al.1999). However, the genome-wide mutation reduces the sensitivity ofWsex/Wasextohand allowsWsex/

rate for most eukaryotes is hotly contested. Some

esti-Wasexto be⬎1 with more realistic levels of dominance.


per genome per generation (Lynchet al.1999) while cost of sex over a range of conservative parameter esti-mates. When the effects of recessive deleterious muta-other estimates suggest that it is at least an order of

magnitude less (Keightley and Eyre-Walker 1999, tions are combined with epistasis or other advantages of sexual reproduction (Howard and Lively 1994; 2000). The average value ofsfor spontaneous

deleteri-ous mutation is in the range 0.01–0.1 (Lynch et al. West et al. 1999), the twofold cost of sex could be completely overcome.

1999). Although previous estimates have been higher,

a recent analysis suggests that the average value of h S. Otto and N. Takebayashi provided helpful discussion and

identi-isⵑ0.1 (Garcı´a-Doradoand Caballero2000). It is fied problems in an early version of the model. E. D. Brodie III, C. M. Lively, K. B. Quinlan, B. J. Ridenhour, M. J. Simmons, and two questionable whether any species are truly panmictic

anonymous reviewers provided helpful comments and clarified the (Hastings and Harrison 1994), so f is probably

al-presentation. This work was supported by a fellowship from Natural ways⬎0. For example, the average values offfor insects, Sciences and Engineering Research Council of Canada to A.F.A. includingDrosophila melanogaster, are in the range 0.03–

0.15 (WadeandGoodnight1998). Reportedfvalues are even higher in a variety of other organisms.

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