vol. 168, no. 5 the american naturalist november 2006 ൴

Host-Parasite and Selection on Sex through the Effects of Segregation

Aneil F. Agrawal1,* and Sarah P. Otto2,†

1. Department of Ecology and , University of 1982) and is now commonly called the Red Queen hy- Toronto, 25 Harbord Street, Toronto, Ontario M5S 3G5, Canada; pothesis. This idea is based on the premise that parasites 2. Department of Zoology, University of British Columbia, 6250 adapt to exploit host genotypes and that sex allows hosts University Boulevard, Vancouver, British Columbia V6T 1Z4, Canada to change genotypes rapidly so that they can remain a step Submitted January 21, 2006; Accepted June 30, 2006; ahead of their relentless enemies. Electronically published September 20, 2006 Most of the theoretical work on the subject has involved computer simulations in which an obligately sexual host Online enhancement: appendix. competes against obligately asexual mutants (e.g., Hamilton et al. 1990; Howard and Lively 1994; Galvani et al. 2003). Because these obligately asexual mutants are completely reproductively isolated from their sexual pro- abstract: The advantage of producing novel variation to keep genitors, such models investigate the “group selection” apace of coevolving species has been invoked as a major explanation advantages of sex in that a sexual group competes against for the and maintenance of sex (the Red Queen hypoth- an asexual group. There are numerous examples where esis). Recent theoretical investigations of the Red Queen hypothesis sexual species give rise to fully asexual mutants in which have focused on the effects of recombination in haploid species, finding that species interactions rarely favor the evolution of sex case such models are appropriate. unless selection is strong. Yet by focusing on haploids, these studies However, these models do not address more subtle have ignored a potential advantage of sex in diploids: generating changes in the mode of reproduction. For example, rather novel combinations of alleles at a particular locus through segre- than the extremes of all or none, an organism may produce gation. Here we investigate models of host-parasite coevolution in some fraction j of its offspring sexually and the remainder, diploid species to determine whether the advantages of segregation 1 Ϫ j, asexually, where j can take any value from 0 to 1. might rescue the Red Queen hypothesis as a more general explanation In general, the results of group selectionist models provide for the evolution of sex. We find that the effects of segregation can favor the evolution of sex but only under some models of infection a poor guide for what to expect when reproductive mode and some parameter combinations, almost always requiring inbreed- can evolve via small, gradual steps. This problem is best ing. In all other cases, the effects of segregation on selected loci favor studied using “modifier” models (Feldman et al. 1997), in reductions in the frequency of sex. In cases where segregation and which one tracks the evolutionary fate of a that mod- recombination act in opposite directions, we found that the effects ifies reproductive mode (e.g., a gene that alters j). of segregation dominate as an evolutionary force acting on sex in Following this approach, previous authors have inves- diploids. tigated theoretically whether parasites favor the spread of a gene that increases the amount of sex and/or recom- Keywords: Red Queen hypothesis, host-parasite coevolution, evolu- tion of sex, segregation, modifier model. bination in haploid species (Parker 1994; Peters and Lively 1999; Otto and Nuismer 2004). This work has demon- strated that coevolution with parasites can select for in- The idea that coevolution with parasites might favor the creased levels of sex or recombination but only under evolution of sex has been popular for more than 2 decades specific conditions (e.g., particular genetic architectures of (Jaenike 1978; Bremermann 1980; Hamilton 1980; Bell infection and high virulence). In fact, the most general modifier analysis of the Red Queen suggests that parasites * Corresponding author; e-mail: [email protected]. typically generate selection against sex and recombination † E-mail: [email protected]. (Otto and Nuismer 2004). Am. Nat. 2006. Vol. 168, pp. 617–629. ᭧ 2006 by The University of Chicago. In these haploid models, any advantage of sex must arise 0003-0147/2006/16805-41906$15.00. All rights reserved. from the beneficial consequences of recombination. How- 618 The American Naturalist

Table 1: Matching alleles model genetic associations at a locus have been built up by past Host Parasite genotype generations of inbreeding (not selection), and these genetic genotype AA Aa Aa associations reduce fitness. Consequently, segregation can p Ϫ p p be advantageous because it brings descendants closer to AA wHAAPAA 1 v wHAAPAa 1 wHAAPaa 1 p p Ϫ p Ϫ Hardy-Weinberg equilibrium, favoring the evolution of wPAAHAA 1 wPAaHAA 1 u wPaaHAA 1 u p Ϫ p Ϫ p Ϫ sex. Aa wHAaPAA 1 v wHAaPAa 1 v wHAaPaa 1 v p p p Our results allow us to compare the conditions under wPAAHAa 1 wPAaHAa 1 wPaaHAa 1 p p p Ϫ aa wHaaPAA 1 wHaaPAa 1 wHaaPaa 1 v which parasites select for recombination versus segregation p Ϫ p Ϫ p wPAAHaa 1 u wPAaHaa 1 u wPaaHaa 1 and the relative strength of these two effects (see “Dis- Note: of host genotype ij when exposed to parasite genotype mn cussion”). Overall, our results indicate that parasites typ-

(wHijPmn) and fitness of parasite genotype mn when exposed to host genotype ically select against sex in populations of randomly mating ij (wPmnHij). diploid hosts but can select for sex when inbreeding is present and heterozygotes have higher than average fitness ever, in diploids involves two pro- in host-parasite interactions. cesses: recombination and segregation. To appreciate fully how the Red Queen might shape reproductive mode, we Model and Results must also understand whether segregation is advantageous in the presence of parasites. In diploids, segregation breaks We model a single host species interacting with a single down associations at a locus between the alleles on ho- parasite species, where the A locus mediates the interac- mologous chromosomes (measured by a positive or neg- tions and affects fitness, whereas the M (modifier) locus ative inbreeding coefficient). Because such associations do affects the mode of reproduction. Our model is completely not exist in haploids, the potential advantages of segre- deterministic, so there is no benefit to sex from segregation gation were absent in these previous theoretical studies of recreating genotypes that are lost stochastically in finite the Red Queen hypothesis. While the studies above have populations (Galvani et al. 2003; see also Antezana and provided an understanding of how the Red Queen acts on Hudson 1997). recombination, they have said nothing of segregation. Be- The first stage in the life cycle is selection via infection cause sex and recombination are not functionally equiv- by parasites. We assume each diploid host encounters a alent in diploids, haploid Red Queen models provide a single diploid parasite at random. Whether this encounter qualitatively incomplete picture of how parasites affect the results in infection depends on the host’s genotype at the evolution of sex in diploid hosts. A locus and the genotype of the parasite at its correspond- Here we use a modifier approach to identify the con- ing A locus. If an infection occurs, the host’s fitness is ditions under which parasites generate selection for in- reduced byv . The fitness of parasites that are resisted by creased sex in diploid hosts. We study a model in which hosts is reduced by u. Tables 1–3 show the fitness of each only a single locus directly affects fitness (with selection host genotype when interacting with each parasite geno- mediated by parasites) so that any observed advantage to type. We consider three different classes of host-parasite sex must arise through the effects of segregation at a locus interaction models commonly used in Red Queen models rather than recombination among loci. As we shall see, (Otto and Michalakis 1998): a matching alleles (MA) coevolution with parasites can favor the spread of modifier model, an inverse matching alleles (IMA) model, and a alleles that increase the frequency of sex in the host, but gene-for-gene (GFG) model. In the MA model, the host it more often favors modifier alleles that decrease the fre- can detect and then eliminate parasites carrying alleles that quency of sex. Why do host-parasite interactions select against sex in Table 2: Inverse matching alleles model this model of segregation? Hosts that have survived selec- Host Parasite genotype tion have genotypes at the selected locus that tend to resist genotype AA Aa aa parasites more often than expected at Hardy-Weinberg p p p Ϫ AA wHAAPAA 1 wHAAPAa 1 wHAAPaa 1 v equilibrium; these favorable associations are broken down p Ϫ p Ϫ p wPAAHAA 1 u wPAaHAA 1 u wPaaHAA 1 by segregation. That is, segregation (like recombination) p p p Aa wHAaPAA 1 wHAaPAa 1 wHAaPaa 1 breaks apart beneficial combinations of alleles that have p Ϫ p Ϫ p Ϫ wPAAHAa 1 u wPAaHAa 1 u wPaaHAa 1 u been built up by past generations of selection. p Ϫ p p aa wHaaPAA 1 v wHaaPAa 1 wHaaPaa 1 p p Ϫ p Ϫ When can host-parasite interactions favor sex? Higher wPAAHaa 1 wPAaHaa 1 u wPaaHaa 1 u rates of sex tend to evolve when hosts inbreed and when Note: Fitness of host genotype ij when exposed to parasite genotype mn host-parasite interactions cause heterozygotes to be fitter (wHijPmn) and fitness of parasite genotype mn when exposed to host genotype on average than homozygotes. Under these conditions, ij (wPmnHij). Segregation and the Red Queen 619

Table 3: Gene-for-gene model Host Parasite genotype genotype AA Aa aa p Ϫ p Ϫ p Ϫ Ϫ AA wHAAPAA 1 c wHAAPAa 1 c wHAAPaa (1 c)(1 v) p Ϫ p Ϫ p Ϫ wPAAHAA 1 u wPAaHAA 1 u wPaaHAA 1 k p Ϫ p Ϫ p Ϫ Ϫ Aa wHAaPAA 1 c wHAaPAa 1 c wHAaPaa (1 c)(1 v) p Ϫ p Ϫ p Ϫ wPAAHAa 1 u wPAaHAa 1 u wPaaHAa 1 k p Ϫ p Ϫ p Ϫ aa wHaaPAA 1 v wHaaPAa 1 v wHaaPaa 1 v p p p Ϫ wPAAHaa 1 wPAaHaa 1 wPaaHaa 1 k

Note: Fitness of host genotype ij when exposed to parasite genotype mn (wHijPmn) and

fitness of parasite genotype mn when exposed to host genotype ij (wPmnHij). differ from its own; this model is applicable to species with resistance allele, A, such that in the absence of parasites, an capable of recognizing and clearing the fitnesses of the AA, Aa, and aa genotypes are1 Ϫ c , nonself antigens. In the IMA model, each allele in the host 1 Ϫ c, and 1, respectively. Similarly, there is a cost to par- confers the ability to recognize a particular allele in the asites of carrying the infectious allele, a, such that the parasite; infection occurs only if none of the host alleles fitnesses of AA, Aa, and aa genotypes when on susceptible are able to recognize any of the alleles carried by the par- hosts are 1, 1, and1 Ϫ k , respectively. asite. In the GFG model, a host can resist a parasite only What we refer to as the “infectious allele,” a, is often if the host expresses a resistance allele, A, and the parasite referred to in the literature as the “virulent” or “virulence” expresses a noninfectious (avirulent) allele, A; this genetic allele. We prefer the term “infectious allele” because this model is thought to underlie many plant-fungal interac- allele allows its carrier to infect a broader array of host tions (Flor 1956; Crute et al. 1997). genotypes than does the alternative allele. The term “vir- To model host-parasite interactions in diploids, we must ulence” is used in this article to describe the reduction in specify the susceptibility of heterozygous hosts and the host fitness caused by infection. infectiousness of heterozygous parasites (see Nuismer and In the host, there are four types of chromosomes at the Otto 2005). For the sake of brevity, we discuss only the modifier and selected loci: MA, Ma, mA, and ma, which simplest reasonable set of assumptions about the gene ex- we refer to by the indices 1–4, respectively. Let Hij be the pression and fitness patterns in heterozygotes; more gen- frequency at the beginning of a generation of hosts car- eral results are presented in the appendix in the online rying chromosomes i and j, wherei, j ෈ {1,2,3,4} . We edition of the American Naturalist. In the simplest version distinguish between genotype ij and ji in our notation even of the MA model, heterozygous hosts express both alleles though the frequencies of the two genotypes are equal p and so are unable to identify any parasite as being nonself; (HijH ji ) under the mating and selection regimes that consequently, the fitness of heterozygous hosts is always we consider. The frequency of genotype ij after selection (s) p lower than or equal to the fitness of homozygous hosts. isHijHw ij Hij/w H , where wHij is the average fitness of

Conversely, in the simplest version of the IMA model, hosts with genotype ij and wH is the average fitness of all expression of both alleles allows heterozygous hosts to rec- hosts. We calculate the average fitness of a host with ge- p ͸ ognize and eradicate any parasite; consequently, hetero- notype ij aswHijkl Pw kl HijPkl , where Pkl is the frequency zygotes are at least as fit as the best homozygote. Domi- of diploid parasites carrying chromosomes k and l and nance relationships in the GFG model are better informed wHijPkl is the fitness of a host with genotype ij when exposed by empirical data than the other models (Flor 1956; Crute to a parasite with genotype kl. To simplify the presentation, et al. 1997). Resistance alleles are typically dominant to we focus on the evolution of sex in hosts and assume that susceptibility alleles, and noninfectious alleles are typically the M allele is fixed at the modifier locus in parasites dominant to infectious alleles. Thus, in the simplest GFG (analogous results for the evolution of sex in parasites are model, a heterozygous host has the same fitness as the presented for comparison below). Consequently, we need resistant AA homozygote. to keep track of only chromosomes 1 and 2 (carrying It is often thought that resistant alleles in the host (or alleles A and a) in parasites. The frequency of parasite (s) p infectious alleles in the parasite) might be more costly to genotypes after selection is given byPklPw kl Pkl/w P , express than alleles that do not induce a resistance response where wPkl is the average fitness of parasites with genotype

(or do not mount a strong attack on the host). Thus, we kl and wP is the average fitness over all parasites. Specif- p ͸ incorporate two additional parameters in the GFG model. ically,wPklkl Hw ij PklHij , where wPklHij is the fitness of a There is a cost to hosts of carrying and expressing the parasite with genotype kl when exposed to a host with 620 The American Naturalist

( p ͸ (s) Table 4: Selection coefficients for different mod- fori j , wherej¯ ij jijH ij is the proportion of all els of infection to leading order in the virulence, offspring that are produced sexually and gi is the frequency v, and cost of resistance, c of haplotype i in the gamete pool that forms the sexually

Model Parameter produced offspring. The gi are given by (MA)≈ 2 Matching allelessHAAq Av ; (MA) ≈ p sHAa v; g1 (MA)≈ 2 sHaaq av (s)(ϩ s)(ϩ ϩ s)(ϩ s)(Ϫ ϩ s) ≈ Ϫ ϩ j(H11H 12 ) (j hjdj)[H13H 14(1 r) Hr 23 ] FMA v(1 2qqAa) , (2a) (IMA)≈ 2 j¯ Inverse matching allelessHAAq av ; (IMA) ≈ sHAa 0; p g 2 (IMA)≈ 2 sHaaq Av F ≈ v(1 Ϫ 2qq) j(H (s)(ϩ H s)() ϩ (j ϩ h dj)[H s)(ϩ H s)((1 Ϫ r) ϩ Hrs) ] IMA Aa 12 22 j 24 23 14 , (2b) (GFG)≈ ϩ 2 ¯ Gene-for-genesHAAc q av ; j (GFG)≈ ϩ 2 sHAac q av; p (GFG) ≈ g 3 sHaa v ≈ Ϫ 2 Ϫ FGFG v(1 qa) c (j ϩ dj)(H (s)(ϩ H s)() ϩ (j ϩ h dj)[H s)(ϩ H s)((1 Ϫ r) ϩ Hrs) ] 33 34 j 13 23 14 , (2c) j¯ p genotype ij. Because the modifier locus has no direct effect g 4 on fitness, expressions for the wHijPkl and wPklHij terms de- ϩ (s)(ϩ s)(ϩ ϩ s)(ϩ s)(Ϫ ϩ s) (j dj)(H34H 44 ) (j hjdj)[H24H 14(1 r) Hr 23 ] pend only on the A locus. For example,wwH11P11 ,H13P11 , , (2d) j¯ andwH33P11 are all equivalent, because they refer to the fitness of an AA host interacting with an AA parasite; thus, where r is the recombination rate between the M and A we denote this fitness more clearly as wHAAPAA. Fitnesses under the different host-parasite models are given in tables loci. 1–3. Because we have limited ourselves to a single selected The evolution of sex has attracted much attention be- locus, A, selection on reproductive mode will reflect the cause of the ubiquity of sex despite intrinsic costs (e.g., advantages (or disadvantages) of segregation alone. the cost of meiosis; Williams 1975; Charlesworth 1980). Following selection, reproduction occurs. The host can Before attempting to understand how the benefits of sex produce offspring both sexually and asexually. We denote might overcome any intrinsic costs, we must first establish the fraction of sexually produced offspring by genotype ij whether sex is advantageous in the absence of such costs. If modifiers increasing the frequency of sex are unable to as j . This fraction is determined by the modifier locus, ij spread in the absence of costs, they will be unable to spread M. Individuals with genotype MM produce a fraction j p p p with a cost. Having identified conditions where increased of their offspring sexually (i.e.,j11j 12j 22 j ), while the fraction of sexually produced offspring in Mm sex can evolve, we then add an intrinsic cost of sex to ϩ ϩ equations (1) to determine the extent to which the ad- and mm individuals isj hjdj and j dj , respectively. For the sake of discussion, we will assume that the m allele vantages of segregation outweigh these costs. increases the amount of sex in a directional fashion (i.e., For simplicity, we assume that parasites reproduce sex- 1 ! ! ually via random mating: dj 00andhj 1 ), although the results apply generally to any type of modifier. Among offspring produced sex- P (r)(p (P s)(ϩ P s)2), (3a) ually, a fraction f are the result of gametophytic selfing 11 11 12 (i.e., selfing among the genetically identical gametes pro- (r)(p s)(ϩ s)(s)(ϩ s) P12(P 11P 12)(P 12P 22 ), (3b) duced by a haploid parent), and the remainder are pro- (r)(p s)(ϩ s)2 duced by random union of gametes. We use gametophytic P22(P 12P 22 ). (3c) selfing as a proxy for the effects of inbreeding by any mechanism, including population subdivision, though we We also examined models in which parasites could self or are aware that the effects are not exactly equivalent (see were haploid. The results (not shown) were qualitatively “Discussion”). similar to our findings for diploid, randomly mating Host genotype frequencies after reproduction are given parasites. by As an alterative to describing the dynamics of genotype frequencies within the host population, we can describe (r)(p Ϫ s)2ϩ Ϫ ϩ the dynamics of the allele frequencies and the patterns of Hii(1 j ii)H iij¯[g i(1 f ) gf i ], (1a) associations among these alleles. In doing so, we use the (r)(p Ϫ s) ϩ Ϫ Hij(1 j ij)H ijj¯[gg i j(1 f )] (1b) following variables: pM, pA,,,,,CCCCM, MA, AMA, ∅ M, A Segregation and the Red Queen 621

CCMA, MMA,, AMA , and C, MA , where pi is the frequency of pected fitness of host genotypes AA, Aa, and aa as p Ϫ p Ϫ p Ϫ allele i andCS, T is the association between the set of alleles wHAA1 sw HAA, HAa1 sw HAa , and Haa1 s Haa , in S on one chromosome with the set of alleles in T on where the selection coefficients, si, depend on the model the homologous chromosome. These associations are mea- of infection, are of order y, and are given in table 4. At ˆˆ 2 sured as central moments following Barton and Turelli QLE,CCM, AMA and, ∅ are 0 to O(y ) . The leading order (1991): terms in the remaining associations are

1 Ϫ j Cˆ p pp fϪ ppF ϩ O(y2), (4a) p ͸͹Ϫ ͹ Ϫ A,AAa()j Aa CS, TijijxijxH [U (x) p ][V (x) p ], ij ({x෈Sx }{෈T }) dj Cˆ p ppfϩ pp(2h Ϫ 1) ϩ O(y2), (4b) M,MMmMm[]j j where Uij(x) is an indicator variable that is unity if ge- notype ij carries allele x on the chromosome represented 1 ˆ p Ϫ Ϫ ϩ ϩ 2 CMA,MMmAaHAaHAAApppp(1 j)(s s p F)f O(y ), (4c) by i and is 0 otherwise. The function Vij(x) is defined j analogously but for the homologous chromosome repre- dj sented by j. These association measures also have straight- Cˆ p Ϫ pppp22[hp ϩ (1 Ϫ h )p ]F MA,AMmAaj 2 j M j m forward interpretations. For example,CA, A is the covari- ance between alleles at the A locus within individuals and dj ϩ pppp[hp ϩ (1 Ϫ h )p ]f ϩ O(y3), (4d) is 0 when the population is at Hardy-Weinberg equilib- j MmAa j m j M rium. The functionCMA, ∅ is the covariance between the ˆ p ϩ 2 M and A alleles on the same chromosome (i.e., the gametic CMA,MAfp M p m p A p a O(y ), (4e) disequilibrium). The functionC is the three-way as- MA, A p Ϫ sociation between the M allele and homozygosity at the where F is a measure of dominance: F 2wHAa ϩ A locus, which is positive if the M allele tends to be found (wHAAw Haa). For each of the models of infection genetics in homozygotes at the A locus more often than the m (MA, IMA, and GFG), F is given in table 4 and is of order y. allele. As we will see,CCA, AMA and, A are of special im- portance in understanding the evolution of sex in this Using these QLE associations, we can calculate changes model. It is useful to describe the population in terms of in allele frequencies at the A and M loci. The change in allele frequencies and association measures, because the M frequency of the A allele in the host over a generation is allele evolves through its effects on the genetic associations. ≈ Ϫ ϩ Ϫ ϩ 2 DpAAaAHAaHAAaHaaHAapp[p (s s ) p (s s )] O(y ). (5) Analogously for parasites, we use qA andDA, A to represent the frequency of the A allele and the covariance of A alleles within individuals. As expected, the sign and magnitude of the change in the frequency of the A allele depends on the distribution of parasite genotypes and the selection that they induce on Analytical Approximations the host. The change in frequency of the host modifier allele, m, To make analytical progress, we assume selection is weak over a generation is relative to the amount of genetic mixing (segregation and recombination). Doing so allows us to employ a separation Dp p Cˆ F ϩ O(y4) (6a) of timescales known as the quasi-linkage equilibrium mMA,A (QLE; Kimura 1965; Nagylaki 1993), in which the genetic dj dj p Ϫ ppppJ22F 2ϩ ppppJfF ϩ O(y 4), (6b) associations reach their steady state faster than the alleles jj2 MmAa1 MmAa2 change in frequency. Though the assumptions underlying p ϩ Ϫ p Ϫ ϩ the QLE may seem counter to the idea of coevolution, this whereJ1 hpj M (1 hj)pJm and 2 (1 hj)pM technique has been shown to work well (Otto and Nuismer hpj m. Equation (6a) indicates that the modifier evolves in

2004), provided that selection is weak per locus, as it will response to the three-way associationCMA, A , which arises be when virulence is low, resistance is weakly genetically as the modifier alters the intralocus association,CA, A (the based, and/or many loci mediate host-parasite interac- departure from Hardy-Weinberg at the selected locus A). tions. The steady state (or QLE) associations are denoted The selective consequences of alteringCA, A depend on byCˆ and are calculated using the recursions above under whether heterozygotes are more fit (F 1 0 ) or less fit the assumption thatv , u, c, k, f, and dj are weak, that is, (F ! 0 ) than expected on the basis of the fitness of the of order y, where y is some small term. homozygotes. The first term in equation (6b) is always Assuming weak selection, we can approximate the ex- negative, indicating selection against sex. The second term 622 The American Naturalist can be positive (i.e., favor sex), but only if there is in- Because the costs of sex are thought to be large, but we breeding (f 1 0 ) and if heterozygotes are more fit than the have assumed weak selection at locus A, it would be im- average of the two homozygotes (F 1 0 ). As inbreeding possible for a single gene to pay for the entire costs of sex. increases, the second term dominates equation (6b); al- Thus, to determine whether sex can evolve in the face of tering the QLE analysis to allow for high rates of inbreed- such costs, we must scale our results up to the total effect ing demonstrates that equation (6b) remains a reasonable of selection across a genome. We now allow there to be n approximation, provided that f is replaced byf(1 Ϫ f ) . independently selected loci rather than just one. For ex- Equation (6b) assumes that f and F are not both very ample, this model could describe a host species under small (i.e., of order y2); if they are, additional terms must attack by n different parasite species in which resistance be considered to predict the fate of a modifier (see the to each parasite is controlled by a single locus specific to one-species analysis of Otto 2003). that parasite. Alternatively, resistance to a single parasite To evaluate whether the Red Queen can select for sex, could be under multilocus control, but the effect of each we must determine whether F is positive. For the MA locus on the probability of infection is independent of the ≈ Ϫ ϩ model, we find thatFMA v(1 2qqAa) , which is always other loci. With these changes to the model, and assuming negative; thus, sex is never favored in hosts. This result that n isO(1/y) , equations (6) become makes sense because heterozygotes can be infected by all parasites and are least fit, but segregation restores hetero- dj 1 n Dp p Ϫ ppJ͸ p222(1 Ϫ p ) F zygosity in an inbred host population. In the IMA model, mMm2 1 iii j 1 Ϫ b []i ≈ Ϫ FIMA v(1 2qqAa), which is always positive; thus, the dj1 ϩ b n Red Queen can favor more sex, provided there is sufficient ϩ ppJf͸ p(1 Ϫ p ) Ϫ Mm2 iiiF (7) inbreeding. This result again makes sense because segre- j 1 b []i gation restores heterozygosity in inbred hosts, and het- bb Ϫ djppJ1 ϩ djp [1 ϩ (2h Ϫ 1)p ] erozygotes can resist all parasites and are most fit. In the 1 Ϫ bj Mm1{}1 Ϫ bj m j M ≈ Ϫ 2 Ϫ 1 GFG model,FGFG v(1 qa) c . Becausev c is re- ϩ O(y3), quired to maintain a , the sign of FGFG changes from negative to positive depending on the fre- quency of the infectious parasite allele a. Whether these where pi is the allele frequency at the ith locus and Fi is cycles favor or disfavor sex overall is explored via simu- the dominance measure for fitness at that locus. The first lation in the next section. two terms are similar to those in equation (6b). They These results apply only to the simplest version of each reflect the indirect selection that arises because of the as- model of infection (tables 1–3). More general results are sociations that the modifier develops with selected loci. As presented in the appendix, where we show that the value in equation (6b), the first term is always negative, but the of F depends on the assumptions made about the sus- second term can be positive. The last term shows the de- ceptibility of heterozygous hosts and the infectiousness of crease in the modifier due to the intrinsic cost of sex, that heterozygous parasites. is, direct selection against the modifier. If the intrinsic cost In the above, we have assumed no intrinsic costs of sex. of sex is large, then this cost will overwhelm the indirect We have shown above that, under the right circumstances, selection the modifier experiences through its association the modifier can increase in frequency, demonstrating that with any one selected locus. If there are a sufficient number indirect selection can favor sex, but it remains unclear of selected loci, and if inbreeding is present and hetero- whether these indirect benefits are sufficient to overcome zygotes are more fit on average than homozygotes, then intrinsic costs. Costs of sex can be added to the model by the summed indirect benefits could offset the intrinsic cost reducing the number of sexually produced offspring by an of sex. amount b relative to the number of asexually produced Although our focus is on hosts, we can also consider offspring of the same genotype (b p 1/2 represents a two- the evolution of sex in parasites by evaluating the anal- fold cost of sex). Mathematically, the cost of sex is incor- ogous measure of dominance in parasites, PF. In short, porated by multiplyingj¯ in equations (1) by1 Ϫ b and parasitic sex is never favored in the MA or IMA models, p Ϫ Ϫ renormalizing the set of equations (dividing by their sum because the dominance measure, PFMA, PIMAF u(1 so that they remain frequencies). This cost of sex reflects HAa), is always negative (HAa is the frequency of Aa hosts). the reduced efficiency of sexual reproduction as a result This reflects the fact that heterozygous parasites are poor of the costs of searching for and courting a mate, the risks mimics of homozygous hosts (MA model) and have twice associated with mating, the reduction in number of off- as many antigens that the host might recognize (in the spring produced by a sexual couple versus two asexual IMA model), causing heterozygotes to be less fit than the individuals, and so on. average fitness of homozygotes. Sex can be favored because Segregation and the Red Queen 623

Figure 1: Simulation results for inverse matching alleles model. Dark shading indicates cases where heterozygotes were more fit, on average, than homozygotes (F 1 0 ). Striped cells indicate that selection favored the spread of a modifier increasing the frequency of sex. Selection for sex was observed only when there was some inbreeding and selection was not too strong. of the advantages of segregation in the GFG model but scribed above. The change in the frequency of the modifier only during those periods of time when aa hosts are suf- during generations 15,001–20,000 was calculated. To sur- p Ϫ Ϫ ϩ ficiently common, so thatPGFGF u(1 Haa) k be- vey parameter space, we simulated all combinations of the comes positive. following parameter values: (i) baseline frequency of sex, j p 0.05, 0.50, 0.95; (ii) dominance of the modifier, p p hj 0.5; (iii) effect of the modifier,dj 0.02 ; (iv) in- Computer Simulations breeding level,f p 0 , 0.01, 0.10; (v) effect on host fitness p The analytical approximations presented above assume of infection,v 0.01 , 0.1, 0.5, 1.0; (vi) effect on parasite p that selection is weak relative to the amount of genetic fitness of failing to infect,u 0.1 , 1.0; and (vii) costs of p mixing. To examine the evolution of the modifier in hosts resistance and virulence in the GFG model,c 0.1v , over a broader set of conditions, we performed computer 0.9v;,0.9k p 0.1u u. For each unique set of parameter simulations in which we iterated recursions (1)–(3). In the combinations, 10 replicate simulations were performed following, we again ignore the costs of sex and focus on from different initial allele frequencies. Typically, results a single selected locus (b p 0 ,n p 1 ). In the simulations, were very similar across replicate simulations. we allowed the A locus in both hosts and parasites to As predicted by our analytical results, the modifier de- mutate at ratem p 10Ϫ4 to prevent alleles from fixing dur- clined in frequency for all parameters under the MA model ing coevolutionary oscillations. For each run, the initial (results not shown). Under the IMA model, the modifier value for pm was 50%, whereas values for pA and qA were increased in frequency, provided that there was sufficient chosen at random. Initial genotype frequencies were cal- inbreeding (fig. 1). The analytical approximation (eq. [6b]) culated from these allele frequencies assuming no initial accurately predicts the level of inbreeding required to favor genetic associations (i.e., loci were at Hardy-Weinberg with sex when selection is weak, but the approximation worsens no linkage disequilibrium). We used a “burn-in” period as selection becomes strong relative to the amount of sex of 15,000 generations to allow the system to approach its (table 5). Interestingly, increasing the strength of selection long-term dynamics; during this period, the modifier allele in hosts made it harder, not easier, for sex to evolve, in had no effect on reproductive mode. Starting in generation contrast to results from haploid models that focus on the 15,001, the modifier increased the amount of sex as de- effects of recombination (Peters and Lively 1999; Otto and 624 The American Naturalist

Table 5: Comparison of analytical and simulation results for gation) alter genetic associations among alleles and the the inverse matching alleles model of infection selective consequences of doing so. Virulence Baseline amount of sex (j) We built an explicitly coevolutionary model to examine ()v .05 .50 .95 whether a modifier that increased the amount of sex would ∗ p ∗ p ∗ p be favored in the presence of parasites. The clearest result .01 fA .026 fA .003 fA .001 ! ∗ ! ! ∗ ! ! ∗ ! is that, in the absence of inbreeding, parasites typically .03 fS .04 .003 fS .004 .001 fS .002 ∗ p ∗ p ∗ p select against hosts in the models we examined. This result .10 fA .5 fA .026 fA .013 ! ∗ ! ! ∗ ! NP .03 fS .04 .02 fS .03 was true both when selection was weak (analytical results) ∗ p ∗ p .25 NP fA .067 fA .034 and when selection was strong (simulations). Conse- ! ∗ ! ! ∗ ! NP .15 fS .16 .05 fS .06 quently, parasites favor sex less often in our single-locus ∗ p ∗ p .50 NP fA .146 fA .071 diploid models than in analogous two-locus haploid mod- ! ∗ ! NP NP .29 fS .30 els when mating is random (Otto and Nuismer 2004). We 1.00 NP f ∗ p .5 f ∗ p .156 A A discuss the effects of segregation on the evolution of sex NP NP NP and compare these effects to those of recombination. Note: Each cell gives the minimum level of inbreeding required to favor In both the single-species model studied by Otto (2003) an increase in the amount of sex as predicted by the analytical approximation ∗∗ as well as the results presented here, the effects of segre- (ffAS ) and by computer simulations ( ). NP indicates that it is not possible ∗ gation on the evolution of sex depend on whether the to favor sex under any level of inbreeding. In evaluatingfA , we used equation (6b) with f replaced byf(1 Ϫ f ) , which provides a better approximation fitness of heterozygotes is greater or less than the average p p ∗ when f is large. We assume thatpAAq 0.5 in evaluatingf A because of the two homozygotes (i.e., the dominance factor,F ). allele frequencies oscillate around the midpoint in the simulations. We also In Otto’s model, the fitnesses of the three genotypes at p assumepM 0.5 because simulations were initiated with this allele fre- ∗∗ locus A were constants rather than dynamical functions quency. As expected,ffAS is close to under the assumptions of the analysis; that is, selection is weak relative to the amount of genetic mixing (top right of parasite genotype frequencies. That these different mod- of table). els give analogous results is a reflection of the fact that, from the perspective of the modifier, only the pattern of selection in the focal species is important, not the agent Nuismer 2004). Simulation results from the GFG model are of selection (parasites or otherwise). Nevertheless, para- shown in figure 2. Selection favoring sex usually involved sites may be important agents of selection if immunity some inbreeding, positive F (i.e., higher average fitness GFG loci have particular dominance properties with respect to of heterozygotes than homozygotes), low to intermediate fitness. levels of virulence, and relatively high costs of infectiousness. It is therefore necessary to consider dominance in each Increased sex was favored in some cases when F fluc- GFG of the models of infection investigated here. In the simplest tuated in sign over time but never when F was consis- GFG version of the MA model, a heterozygous host is incapable tently negative, as expected from equation (6b). Because of detecting and clearing any parasite because no parasite F ≈ v(1 Ϫ q 2) Ϫ c, the sign of F depends on the fre- GFG a GFG genotype carries an allele that does not match at least one quency of the infectious allele, q , and hence on the coevo- a of the heterozygous host’s alleles. Heterozygous hosts can lutionary dynamics of host and parasite. In particular, low be infected by any parasite genotype, whereas each ho- costs of infectiousness, k, tend to elevate the frequency of mozygous host is susceptible to only a single parasite ge- the infectious allele, q , leading to a negative F and se- a GFG notype. Consequently, the heterozygous host can never be lection against sex for most parameter combinations. In more fit than either homozygote, soF is always negative. contrast, the results appeared less sensitive to the cost of MA In the simplest version of the IMA model, a heterozygous resistance, c. host can recognize at least one allele carried by any of the parasite genotypes. Heterozygous hosts can thus resist all Discussion parasite genotypes, whereas each homozygous host is sus- ceptible to a subset of parasite genotypes. As a result, the Sexual reproduction does not alter the frequency of alleles heterozygous host is never less fit than either homozygote,

(assuming Mendelian inheritance), but it does change their soFIMA is always positive. In the simplest version of the patterns of association. In particular, recombination re- GFG model, the A allele is assumed to be completely dom- duces the disequilibrium between alleles at different loci inant so that the heterozygous host Aa has the same fitness on a chromosome. Similarly, segregation reduces associ- as the AA host. As a result, the heterozygous genotype will ations between alleles at the same locus on homologous be more fit than the average of the two homozygous ge- 1 chromosomes. From a population-genetics perspective, notypes (FGFG 0 ) when the AA homozygote is more fit 1 understanding the advantages of sex requires understand- than the aa homozygote (wAAw aa ). Conversely, the het- ! 1 ing how sexual processes (i.e., recombination and segre- erozygote will be relatively unfit (FGFG 0 ) when waa Segregation and the Red Queen 625

Figure 2: Simulation results for gene-for-gene model. Dark shading indicates cases where heterozygotes were always more fit, on average, than homozygotes (F 1 0 ); light shading indicates fluctuations in the sign of F. No shading indicates that F was always negative. Striped cells indicate that selection favored the spread of a modifier increasing the frequency of sex. The cell marked IC is the only part of parameter space in which the fate of the modifier appeared to depend on initial conditions. Selection for sex occurred only if (i) there was some inbreeding (f 1 0 ) and heterozygotes were relatively fit (F 1 0 ) or (ii) heterozygotes fluctuated between being relatively fit and unfit. These conditions were necessary but not sufficient for sex to be favored. wAA. Our simulations indicate thatFGFG tends to be positive Bearing these dominance attributes in mind, we first when the cost of infectiousness, k, is high (dark shading consider our results when mating is random. When se- in fig. 2). Note that it is possible forFGFG to fluctuate in lection is weak, our QLE approximation given by equations sign, and such fluctuations are observed in some simu- (6) (settingf p 0 ) shows that parasites always select lations (light shading in fig. 2). against segregation (see also Otto 2003). In the absence 626 The American Naturalist of other factors (e.g., inbreeding), the sign of the genetic asite. In Hamilton’s model, the dominance measure, F, associationCA, A is determined only by selection. Sex is can switch signs over time when selection is strong. Ex- detrimental because it reduces the association created by amining his simulations in more detail, we confirmed that selection. For example, if heterozygotes are relatively fit the advantage of sex that he observed was generated by (F 1 0 ), selection by parasites causes an excess of hetero- rapid switches in the form of dominance, causing a mod- ! zygotes (CA, A 0 ). Sex reduces this excess by producing ifier allele to become associated with departures from homozygous offspring from these heterozygous parents Hardy-Weinberg that were, on average, more fit in the (,Aa # Aa r AA Aa, aa), which reduces fitness because next generation. homozygotes are relatively unfit whenF 1 0 . If hetero- In summary, it is difficult to identify conditions under zygotes are relatively unfit (F ! 0 ), selection causes an ex- which parasites favor segregation in the absence of in- 1 cess of homozygotes (CA, A 0 ). Sex reduces this excess by breeding; only in rare cases does F fluctuate sufficiently producing heterozygous offspring from these homozygous rapidly. Under most conditions, segregation is disfavored. parents (AA # aa r Aa ), which again reduces fitness be- This is because segregation reduces the genetic association ! cause heterozygotes are relatively unfit (F 0 ). Put simply, CA, A that was built solely by selection. When inbreeding selection creates an excess of good allele combinations; sex occurs, selection is no longer the only force affecting ge- destroys this excess. netic associations. As noted by Bennett and Binet (1956), The above logic and the QLE analysis on which it is inbreeding creates not only an excess of homozygotes but based work well, provided that the sign of F remains con- also an excess of the four double homozygotes: MMAA, stant. When parasites have low virulence, coevolutionary MMaa, mmAA, and mmaa. Because mm individuals are dynamics are slow, and we can think of selection as being more likely to engage in sex than MM individuals, the approximately constant over short timescales. Although crossmmAA # mmaa occurs more often than expected, weak selection justifies the use of the QLE approximation, given the genotype frequencies and the average frequency strong selection is also of interest. Parasites have been of sex. Conversely, MMAA and MMaa individuals are less invoked in discussions of sex as a potentially important likely to engage in sexual reproduction with one another. agent of selection because the pattern of selection they When heterozygotes are more fit (F 1 0 ), the cross generate can change rapidly over time when virulence is mmAA # mmaa is beneficial to the m allele because it strong. If parasites are virulent and coevolutionary dy- results in heterozygous offspring at the fitness locus namics are fast, the dominance parameter F could change (mmAa). This explains the second term in equation (6b), signs over short timescales. which favors the spread of the m allele as long as inbreed- Imagine that heterozygotes are initially relatively fit ing is present and the form of host-parasite interactions (F 1 0 ) so that selection by parasites creates an excess of causes the fitness of heterozygotes to be greater than the ! 1 heterozygotes (CA, A 0 ). Sex reduces this excess by con- average fitness of homozygotes (F 0 ). If the extent of verting heterozygotes into homozygotes. As previously dis- inbreeding, f, is too small relative to the degree of dom- cussed, this conversion reduces fitness if the sign of F has inance, F, then the negative effect of random mating de- not changed (i.e., if heterozygotes are relatively fit in the stroying the genetic associations built up by selection (i.e., next generation). However, this conversion is beneficial if first term of eq. [6b]) outweighs the advantage gained the sign of F has changed. This logic implies that selection through generating Aa heterozygotes from the excess of for segregation may occur if F fluctuates sufficiently rap- double homozygotes (i.e., second term of eq. [6b]). In- idly. However, we have little evidence for such fluctuations breeding has similar effects in a single-species model of in the sign of F. The sign of F never changes in the simplest segregation (Otto 2003). With respect to the models ex- versions of MA and IMA models (FMA is always negative, plored here, selection for sex does not occur in the MA and FIMA is always positive). The GFG model is capable model even with inbreeding because the fitness of hetero- of generating fluctuations in F, but this did not commonly zygotes is relatively low (FMA is always negative). Sex can occur in our simulations (fig. 2). In those cases where be favored when there is sufficient inbreeding in the IMA increased sex was favored in the absence of inbreeding (see model because heterozygotes are relatively fit (FIMA is al- fig. 2;v p 0.5 ,k p 0.9u ,j p 0.05 ), fluctuations in the ways positive) and in the GFG model under conditions sign of F were observed, suggesting that there can, under where FGFG is predominantly positive. When parasites certain circumstances, be a short-term advantage to cause hosts to experience overdominance for fitness, as in segregation. the IMA model, our results are similar to results from a As a historical aside, cycles favoring sex were observed single-species model with overdominance (Dolgin and by Hamilton (1980), who used a one-species diploid model Otto 2003). to describe the case where heterozygotes were a unique Both the analytical and simulation results indicate an genotype that could be attacked only by a specialized par- important role for inbreeding. Although we have modeled Segregation and the Red Queen 627 inbreeding as gametophytic selfing (i.e., syngamy of iden- selection is strong in the MA and IMA models. In these tical gametes), we suspect that these results will hold for models, host-parasite coevolution causes rapid fluctua- some other types of inbreeding with only minor modifi- tions in the sign of the relevant nonlinear component of cations. For example, we investigated sporophytic selfing fitness (epistasis), resulting in a short-term benefit to re- (i.e., syngamy among independently produced gametes of combination (Peters and Lively 1999). In our diploid MA a diploid parent) and found that the results are changed and IMA models, the relevant nonlinear component of only by a factor of1 Ϫ 2r(1 Ϫ r) in the second term of fitness (dominance, as represented by F) does not fluctuate equation (6b). The results remain qualitatively the same in sign, so there is no comparable advantage to segregation. if parasites also inbreed (results not shown). What we do In the GFG model, the advantages of recombination can not know, however, is how the model behaves if the in- favor the evolution of sex when selection is sufficiently breeding level of parasites and the inbreeding level of the strong (Parker 1994; Otto and Nuismer 2004), but such hosts they attack are correlated. Such correlations could cases are less common than in the MA and IMA models. arise in spatially structured populations, especially when Like epistasis, dominance can fluctuate rapidly in sign un- the dispersal distances of hosts and parasites are similar. der some conditions in the GFG model, in which case Our study of the Red Queen and segregation allows for segregation can also provide an advantage to sex. Overall, a comparison with previous studies that have focused on it appears that strong selection does not generate an ad- recombination (table 6). We begin this comparison as- vantage to segregation as often as it generates an advantage suming random mating. Otto and Nuismer (2004) ana- to recombination, because dominance (F) is less likely to lyzed an explicitly coevolutionary haploid model to study vary in sign over time than epistasis. the evolution of recombination under the Red Queen. In diploids, a modifier that increases the amount of sex Similar to our results with respect to segregation, they simultaneously increases both segregation and recombi- found that none of the models of infection (MA, IMA, or nation. Previous studies (Parker 1994; Peters and Lively GFG) favored recombination under the assumptions of 1999; Otto and Nuismer 2004) that have examined the weak selection and random mating. In both our model Red Queen in haploid hosts have focused exclusively on and theirs, the sexual process of interest (segregation or the effects of recombination. Because segregation is almost recombination) reduces a genetic association (homozy- always disadvantageous when mating is random, haploid gosity or gametic disequilibrium, respectively) built by se- models will tend to overestimate how often parasites select lection. Selection causes host genotypes that are better able for sex in diploid hosts. For example, consider a diploid to resist the current population of parasites to become MA model involving two immunity loci. On the basis of more common; breaking apart these favorable host ge- the two-locus haploid model, we would predict that the notypes through sex produces offspring that are, on av- advantages of recombination would favor sex in the pres- erage, more susceptible to attack. ence of highly virulent parasites, but on the basis of the While the recombination and segregation results are single-locus diploid model, we would predict that the dis- very similar under weak selection and random mating, advantages of segregation would select against sex for all differences arise when selection is strong. The advantages levels of virulence. In simulations of a two-locus diploid of recombination can favor the evolution of sex when MA model (not shown), we found that a modifier that

Table 6: Heuristic guide to the results of Red Queen models Infection model Advantages to sex through recombination Advantages to sex through segregation Weak selection: MA No No IMA No Yes, but requires inbreeding GFG No Yes, but requires inbreeding and high costs of infectiousness Strong selection: MA Yes No IMA Yes No GFG Possible, depends on strength of selection and costs Yes, but usually less often than with weak selection of resistance and infectiousness Note:MA p matching alleles, IMA p inverse matching alleles, GFG p gene -for-gene. Recombination results summarized from Parker (1994), Peters and Lively (1999), and Otto and Nuismer (2004), all of which assumed haploid hosts and parasites as well as random mating. Inbreeding may create benefits to recombination under some models of infection, but this has not been studied to date. Weak/strong selection is defined relative to the rates of sex and recombination within the population. Note that in a haploid two-locus, two-allele model, MA and IMA interactions generate equivalent results. 628 The American Naturalist increased sex always declined in frequency even under high often prevail, narrowing the conditions under which sex virulence. These results imply that the negative effects of is favored in diploids relative to haploids. Only if there is segregation outweigh the positive effects of recombination. significant inbreeding does segregation appear likely to Our comparison of segregation and recombination has increase the possibility for sex to evolve. Of course, it so far assumed no inbreeding. We found that segregation remains possible that the models ignore key aspects of the can be advantageous in the presence of inbreeding in the biology that are critical to understanding the Red Queen, IMA model and, under some conditions, in the GFG such as finite population size (Martin et al. 2006) or non- model, because segregation results in the production of random transmission (Agrawal 2006). If so, then the Red heterozygotes, which are more fit, on average, than ho- Queen may be more important than current models sug- mozygotes under these models and yet are relatively in- gest, but the reasons will be more complicated than orig- frequent because of inbreeding. To date, all of the Red inally believed. Queen models investigating the effects of recombination between selected loci have assumed random mating (Par- Acknowledgments ker 1994; Peters and Lively 1999; Otto and Nuismer 2004). A recent one-species model (Roze and Lenormand 2005) We thank two anonymous reviewers for helpful sugges- demonstrated that sporophytic selfing (and presumably tions. This work was supported by Discovery Grants to population structure) can create an advantage to increased A.F.A. and S.P.O. from the Natural Sciences and Engi- recombination if there is negative dominance # neering Research Council of Canada. dominance (d # d ) epistasis. Host-parasite interactions might be a common source of such epistasis. Indeed, dip- loid versions of the MA, IMA, and GFG models with two Literature Cited # immunity loci (A and B) generate negatived d epistasis Agrawal, A. F. 2006. Similarity selection and the evolution of sex: at QLE (A. F. Agrawal and S. P. Otto, unpublished data). revisiting the Red Queen. PLoS Biology 4:1364–1371. This suggests that increased rates of recombination would Antezana, M. A., and R. R. Hudson. 1997. 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Heredity 10:51– vored, because the advantage to recombination arises from 55. a difference in fitness between two types of sexually pro- Bremermann, H. J. 1980. Sex and polymorphism as strategies in host- duced offspring (i.e., selfed recombinant offspring vs. pathogen interactions. Journal of Theoretical Biology 87:671–702. selfed nonrecombinant offspring) rather than a difference Charlesworth, B. 1980. The cost of sex in relation to mating system. Journal of Theoretical Biology 84:655–671. between sexually versus asexually produced offspring. Sim- Crute, I. R., E. B. Holub, and J. J. Burdon. 1997. The gene-for-gene ulations performed using a diploid model with two im- relationship in plant-parasite interactions. CAB International, New munity loci and sporophytic selfing (not shown) did not York. differ qualitatively from the results described above with Dolgin, E. S., and S. P. Otto. 2003. Segregation and the evolution of one immunity locus and gametophytic selfing. 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