The Effective Size of Populations Infected with Cytoplasmic Sex-Ratio Distorters

The Effective Size of Populations Infected With Cytoplasmic Sex-Ratio Distorters

Jan Engelsta¨dter1 Institute of Integrative Biology, ETH Zurich, Switzerland Manuscript received April 19, 2010 Accepted for publication June 22, 2010

ABSTRACT Many arthropod species are infected with maternally inherited endosymbionts that induce a shift in the sex ratio of their hosts by feminizing or killing males (cytoplasmic sex-ratio distorters, or SRDs). These endosymbionts can have profound impacts on evolutionary processes of their hosts. Here, I derive analytical expressions for the coalescent effective size Ne of populations that are infected with SRDs. Irrespective of the type of SRD, Ne for mitochondrial genes is given by the number of infected females. For nuclear genes, the effective population size generally decreases with increasing prevalence of the SRD and can be considerably lower than the actual size of the population. For example, with male-killing bacteria that have near perfect maternal transmission, Ne is reduced by a factor that is given to a good approximation by the proportion of uninfected individuals in the population. The formulae derived here also yield the effective size of populations infected with mutualistic endosymbionts or maternally inherited bacteria that induce cytoplasmic incompatibility, although in these cases, the reduction in Ne is expected to be less severe than for cytoplasmic SRDs.

IMPLE null models are essential in science. In this is the most useful definition for Ne because ‘‘the S population genetics, this role is filled by the coalescent essentially embodies all of the information Wright–Fisher model and its retrospective counterpart, that can be found in sampled genetic data’’(Sjo¨din et al. the Kingman coalescent. Both of these models have 2005). More recently, Wakeley and Sargsyan (2009) proven to be immensely useful in spite of the fact that have proposed two extensions of the coalescent effective natural populations usually violate the assumptions population size in which they advocate including a made in these models. The reason for this is that often, mutation parameter in the definition and also allowing the Wright–Fisher model can be rescaled so that it for a nonlinear relationship between Ne and N. behaves in many important respects like a more One frequently encountered feature in natural pop- complex population model. This rescaling is achieved ulations that complicates population genetics is infection through the concept of the effective population size, with maternally inherited endosymbionts. In particular, Ne. Roughly speaking, a complex population model is many arthropod species harbor a great number of said to have a certain Ne if the haploid Wright–Fisher phylogenetically diverse microorganisms that influence model with population size Ne experiences the same their hosts’ biology in different ways (Bourtzis and amount of random genetic drift as the complex model. Miller 2003; Bourtzis and Miller 2006; Bourtzis Reflecting the different ways in which drift can be and Miller 2009). Because of their maternal trans- measured, Ne can be defined in different ways, e.g.,as mission, many of these microorganisms—for example, the inbreeding, the variance, or the coalescent effective the bacteria Wolbachia pipientis and Cardinium hertigii— population size. Different definitions often produce have evolved intricate manipulations of their hosts’ the same value for Ne, but may also yield drastically reproductive system that allows them to spread in a host different numbers (Kimura and Crow 1963). population through exploitation of male hosts (repro- The coalescent effective population size is defined ductive parasitism, reviewed in Engelsta¨dter and through the factor by which time needs to be rescaled in Hurst 2009a). Most manipulations involve a shift in a complex population model to produce the standard the sex ratio of their hosts (both primary and at the coalescent with time scale given by the population size N population level), and the inducing endosymbionts are (Nordborg and Krone 2002). It has been argued that consequently referred to as cytoplasmic sex-ratio distorters (SRDs). In some species, genetic males develop into females if they are infected (‘‘feminization’’: Martin et al. 1973; Rigaud 1997; Bouchon et al. 1998; Hiroki 1Address for correspondence: Institute of Integrative Biology, ETH Zurich, Universita¨tsstr.16, ETH Zentrum, CHN K12.1, CH-8092 Zurich, et al. 2002). More commonly, infected males are killed by Switzerland. E-mail: [email protected] the endosymbionts early in their development (male

Genetics 186: 309–320 (September 2010) 310 Jan Engelsta¨dter

TABLE 1 Empirical examples for cytoplasmic SRDs with parameter estimates

Host SRD xu xi xm References Acrea encedona (butterfly) Wolbachia 0 0.5 0Jiggins et al. (2002) Adalia bipunctatab (Ladybird beetle) Rickettsia 0.076 0.506 0.076 Hurst et al. (1993) Drosophila innubilab Wolbachia 0.013 0.509 0.037 Dyer and Jaenike (2004) Gammarus duebenic (freshwater shrimp) microsporidium 0.127 0.706 0.167 Dunn et al. (1993) Hypolimnas bolinaa (butterfly) Wolbachia 0 0.5 0Dyson et al. (2002) With the exception of the feminizing microsporidia in G. duebeni, all SRDs are male-killing bacteria. a In these two butterfly species, maternal transmission and male-killing penetrance is very close to perfect, so that virtually no sons or uninfected daughters are produced by infected mothers. The fact that the infection has not spread to fixation in these species suggests that the fitness benefit of surviving siblings in a brood is absent or very low, leading to the prediction that infected females produce about as many daughters as uninfected females (xi 0:5). b In these two species, parameter estimates were obtained from the transmission rate and the prevalence (pˆ) reported in the respective references (see Equation 2). In the case of D. innubila, I used the data from the 2002 sampling in Dyer and Jaenike (2004), as this was the largest sample and lay in between the other two samples with regard to infection prevalence. c To calculate the parameter values given for this species in Dunn et al. (1993, Table 1), I assumed equal overall offspring production of infected and uninfected females (as indicated in Dunn et al. 1993) and discarded occasionally produced intersexes. Also note that the primary sex ratio in uninfected G. duebeni is determined by environmental cues and can therefore deviate from 1:1. killing: reviewed in Hurst et al. 2003). The adaptive for both mitochondrial and nuclear genes. The ap- advantage of this strategy is seen in an early fitness boost proach is considerably more general than in the two in the surviving females in a brood, for example, through above-mentioned previous studies in that not only male reduced sibling competition or cannibalism of the dead killing, but also feminizing and even endosymbionts brothers (Hurst 1991; Hurst and Majerus 1993; without sex-ratio distorting activity, are covered. How- Jaenike et al. 2003). Some examples for species infected ever, diploid hosts are assumed throughout this article, with male-killing or feminizing SRDs are given in Table 1. so that the derivation for Ne in populations infected with Finally, in haplodiploid species, unfertilized eggs that parthenogenesis-inducing bacteria is left for future would normally develop into males can be diploidized by investigations. different bacteria so that they develop into females. The spread of such bacteria may lead to completely asexually MODEL AND RESULTS reproducing populations (parthenogenesis induction: reviewed in Huigens and Stouthamer 2003). Infection dynamics: Let us assume a randomly mating Previous theoretical studies indicate that cytoplasmic diploid population of size N. Females can be either SRDs will have a strong impact on evolutionary pro- infected by the cytoplasmic SRD or uninfected, and cesses for both mitochondrial and nuclear host genes. transmission of the SRDs is strictly maternal (no This is essentially because the host population con- paternal or horizontal transmission). Uninfected fe- sists of different classes of individuals (male/female, males are assumed to produce offspring in a 1:1 sex infected/uninfected) with different reproductive suc- ratio. Infected females can produce uninfected daugh- cess. Johnstone and Hurst (1996) showed that genetic ters, infected daughters, and sons, and the relative variation in mtDNA is expected to be strongly reduced numbers of offspring of these three classes are denoted during the spread of male-killing bacteria. After the by xu, xi, and xm, respectively. These numbers are male killers have reached a stable equilibrium in the relative to the offspring number of uninfected females; population, mtDNA variation will recover, but will still i.e., infected females may have the same fecundity as be permanently reduced to a value that approximately uninfected females (xu 1 xi 1 xm ¼ 1), but they may corresponds to the expected variation if the population also produce fewer (xu 1 xi 1 xm , 1) or more consisted only of infected females. In other words, the (xu 1 xi 1 xm . 1) offspring than uninfected females. equilibrium Ne equals approximately the number of This parameterization covers a broad spectrum of infected females in this case. Conversely, for nuclear different types of cytoplasmic SRDs, and some examples host genes, Engelsta¨dter and Hurst (2007) showed for parameters estimated in natural systems are given in through computer simulations that to a good approx- Table 1. imation, a male-killer infected population behaves as if Let us denote by p the proportion of infected females only uninfected individuals were present. among all females in the population. Assuming that the Here, I derive analytical expressions for the coales- host population is large enough relative to selection on cent effective size of host populations infected with SRD spread, so that random genetic drift of the SRD can cytoplasmic SRDs at equilibrium frequency. This is done be ignored, the infection dynamics of the SRD are Ne With Sex-Ratio Distorters 311

TABLE 2 Parameters and auxiliary variables of the model

Parameters N Actual population size xi; xu; xm Numbers of infected daughters, uninfected daughters and sons, respectively, that an infected female produces relative to a total offspring production of 1 of uninfected females Auxiliary variables p; pˆ Frequency and equilibrium frequency, respectively, of the SRD, i.e., proportion of infected females among all females in the population ci; cu; cm Equilibrium proportion of infected females, uninfected females, and males, respectively, among all individuals in the population quu; qui Probability that the mother of an uninfected female is uninfected or infected, respectively (quu 1 qui ¼ 1) qmu; qmi Probability that the mother of a male is uninfected or infected, respectively (qmu 1 qmi ¼ 1)

approximated by a deterministic model with inﬁnite to calculate N e. Let quu be the probability that the population size and follow the equation mother of an uninfected female is also uninfected, and qui the probability that the mother of an uninfected xip p9 ¼ : ð1Þ female is infected. Similarly, let qmu and qmi be the pðxi 1 xuÞ 1 ð1 pÞ=2 probabilities that the mother of a male is uninfected and infected, respectively. We then have Solving p9 ¼ p yields the nontrivial equilibrium fre- cu=2 1 quency of the SRD quu ¼ ¼ ; ð4aÞ cu=2 1 cixu 2xi x 1=2 ˆ i p ¼ : ð2Þ cixu 1 xi 1 xu 1=2 qui ¼ ¼ 1 ; ð4bÞ cu=2 1 cixu 2xi This equilibrium exists and is stable if and only if cu=2 xu x . 1 , i.e., if infected females produce more infected q ¼ ¼ ; ð4cÞ i 2 mu c =2 1 c x ð2x 1Þx 1 x daughters than uninfected females produce daughters. u i m i m u ine This fundamental result goes back to F (1975) and cixm ð2xi 1Þxm has been obtained in a variety of models with different qmi ¼ ¼ : ð4dÞ cu=2 1 cixm ð2xi 1Þxm 1 xu parametrizations (e.g.,Bull 1983; Taylor 1990; Hurst 1991; Lipsitch et al. 1995; Engelsta¨dter and Hurst 2009b). Note that the number of males produced by Since we assume that the population has reached the infected females (xm) does not feature in Equation 2, equilibrium state with regard to SRD infection, these implying that sex-ratio distortion per se is relevant for the probabilities will remain constant in time. Table 2 spread of the SRD only insofar as it increases the number summarizes all parameters and variables of the model. xi of infected daughters or decreases the number xu of Ne for mitochondrial genes: To derive the effective uninfected daughters that infected females produce. population size for mitochondrial genes, we follow the Based on pˆ, we can now derive the equilibrium ancestry of two gene copies backward in time and frequencies cu, ci, and cm of uninfected females, infected determine the average time until coalescence of these females and males among all individuals in the popula- gene copies occurs. We assume that all mitochondrial tion as genomes within a single host are identical (no heteroplasmy). Since males do not transmit mitochondria, we need only to follow the coalescence process in females. ð1 pˆ Þ=2 1 pˆxu c ¼ ; ð3aÞ To this end, we construct a Markov process with the u 1 pˆð1 x x x Þ u i m following four states: (1) both gene copies are in pˆx different uninfected females, (2) both gene copies are c ¼ i ; ð3bÞ i 1 pˆð1 x x x Þ in different infected females, (3) one gene copy is in an u i m infected and the other copy in an uninfected female,

ð1 pˆ Þ=2 1 pˆxm and (4) the two gene copies have coalesced in a single cm ¼ ; ð3cÞ (infected or uninfected) female. Table 3 illustrates these 1 pˆð1 xu xi xmÞ four states and gives the transition probability matrix P with cu 1 ci 1 cm ¼ 1. Finally, these equilibrium fre- between these states (i.e., Pij is the probability of quencies can be used to calculate probabilities of switching from state i in the current to state j in the descent, which will be needed in the following sections previous generation). 312 Jan Engelsta¨dter

TABLE 3 Probability transition matrix P for the ancestral process of mitochondrial genes

2 2 2 1 2 1 quu qui quu 1 qui 1 2quuqui 1 cuN ciN cuN ciN

1 1 01 0 ciN ciN 1 qui 0 qui 1 quu ciN ciN

0001

Solid squares denote the focal gene copy. Open circles denote uninfected females, shaded circle infected females, and the striped shaded circle in the fourth state denotes coalescence in either an infected or uninfected female.

½Nt Applying Mo¨hle’s method of separation of time scales limN /‘ P ¼ P expðtPBPÞ. Here, the rate matrix (Mo¨hle 1998), we can partition the matrix P as PBP is given by P ¼ A 1 B=N , where 0 1 0 1 1 1 2 2 B 0 0 C quu qui 2quuqui 0 B ci ci C B C B C B 01 0 0C B 1 1 C A ¼ @ A ð5aÞ B 0 0 C 0 qui quu 0 B C PBP ¼ B ci ci C: ð7Þ 00 0 1 B C B 1 1 C @ 0 0 A and ci ci 0 1 0000 q2 q2 q2 q2 B uu ui 0 uu 1 ui C B c c c c C B u i u i C B C This shows that by rescaling the coalescence process B 1 1 C B 0 0 C: by a factor ci, the Kingman coalescent is obtained if N is B ¼ B ci ci C ð5bÞ B C sufﬁciently large. Hence, the effective population size B qui qui C @ 0 0 A for mitochondrial genes is given by N e ¼ Nci in this ci ci case, i.e., the number of infected females. This result has 0000 been previously obtained for male-killing bacteria (Johnstone and Hurst 1996) and has important consequences for the interpretation of mitochondrial The matrix A is again a probability transition matrix nucleotide polymorphism data (Hurst and Jiggins and can be interpreted as describing the backward 2005). Figure 1 shows how the scaling factor dynamics on a short time scale. The equilibrium of this 2N =N ¼ c , giving the change in N relative to an short-term process is given by the limit e i e 0 1 uninfected population with N/2 females, depends on 0100 the parameters of the model. N e is substantially reduced B C B 0100C when infected mothers produce only little more in- P :¼ lim An ¼ @ A: ð6Þ n/‘ 0100 fected daughters than uninfected females produce 0001 daughters (xi 0:5) and have a low rate of maternal transmission (relatively large xu), a situation observed in This means that when starting with two gene copies in many male-killing bacteria that are at low infection two different females, the probability that these two frequency within a population. On the other hand, gene copies will be found in two infected females (state when xi is large or xu is low—as occurs in feminizing 2) will converge to one on this short time scale. microbes or male-killers with high maternal transmis- As proven by Mo¨hle (1998), the coalescence process sion rate—N e is increased relative to an uninfected converges to the continuous-time process PðtÞ¼ population. Ne With Sex-Ratio Distorters 313

Kingman coalescent as the population size tends to infinity. The Markov chain for the ancestral process of two gene copies now has the following 10 states: 1. Both genes are in a single uninfected female (but have not coalesced) 2. Both genes are in a single infected female 3. Both genes are in a single male 4. The genes are in different uninfected females 5. The genes are in different infected females 6. The genes are in different males 7. One gene is in an uninfected female and the other in an infected female 8. One gene is in an uninfected female and the other in a male 9. One gene is in an infected female and the other in a male 10. The two genes have coalesced (in an uninfected or igure F 1.—Scaling factor 2N e=N for the effective popula- infected female or in a male) tion size of mitochondrial genes in an SRD-infected population, depending on the relative numbers xi and xu of Table 4 illustrates these 10 states and gives the proba- infected and uninfected daughters that an infected female bility transition matrix P of this Markov process. produces. In this contour plot, the factor by which N e is al- Again, we can separate time scales by partitioning P tered through SRD infection is indicated by areas of different into P ¼ A 1 B=N . The matrices A and B (not shown) color, where the small numbers in the plot give 2N e=N at the n are readily obtained from P. P :¼ limn/‘A takes the borders of these areas. The relative number of sons, xm, that an infected female produces, is assumed to be equal to the form number of uninfected daughters xu. The dashed red line 0 1 marks the situation where infected females produce as many D1 D2 1 1 D1 1 D2 D3 1 D3 B 000 0 C offspring as uninfected females. B Q Q 4 4 Q Q 2 Q C B C B C B ...... C B ...... C P ¼ B C; It should be noted that although the above deriva- B C B D1 D2 1 1 D1 1 D2 D3 1 D3 C B 000 0 C tion is based on a sample size of two, convergence to the @ Q Q 4 4 Q Q 2 Q A Kingman coalescent would also occur for larger sample 000 0 0 0 0 0 0 1 sizes. This was demonstrated by Nordborg and Krone ð8Þ (2002) for the general case of a structured population where the rates of ‘‘migration’’ of ancestral lineages where the formulae for the placeholders D1, D2, D3, and between different ‘‘patches’’ do not depend on the Q are given in the appendix. The matrix P can be population size N (this is the case of a ¼ 0inNordborg interpreted as the short-term equilibrium probability and Krone (2002)). The present model is a special case distribution. Thus, the probability that the two gene of this more general model: patches correspond to copies will be in two different males (state 6), and the infection states (infected or uninfected), and migration probability that the two genes will be in two different 1 corresponds to movement of lineages from the infected females (states 4, 5, and 7) will converge to 4 on this time state to the uninfected state through imperfect maternal scale, whereas the probability that one gene will be in a transmission of the endosymbionts. Thus, the effective female and the other in a male (states 8 and 9) will 1 population size of N e ¼ Nci obtained here is a generally converge to 2 . These probabilities follow from the applicable one, because the structure of the population— simple fact that in spite of the potentially complicated given by the infection states—collapses to the Kingman transmission dynamics and the sex-ratio distortion, coalescent as N tends to infinity. every individual has one maternally and one paternally Ne for nuclear genes: The effective population size derived allele. for nuclear genes will now be derived with the same The scaling factor determining the effective popula- method as in the previous section. In contrast to the case tion size is given by the entries in the last column of the of mitochondrial genes, we now have to also consider rate matrix PBP. The resulting formula for Ne is given in inheritance through males as well as diploidy. Again, we the appendix. This rather unwieldy formula simplifies consider the ancestral process only for a sample of size considerably in a few important special cases that will two, but on the basis of previous results (Mo¨hle 1998; now be considered. Nordborg and Krone 2002), the same process for Special case 1: Infected females produce equal numbers of larger samples is also expected to converge to the uninfected daughters and sons (xm ¼ xu): This situation 1 a Engelsta Jan 314

TABLE 4 Probability transition matrix P for the ancestral process of nuclear genes

000 0 0 0 0 quu qui 0

000 0 0 0 0 0 1 0

000 0 0 0 0 qmu qmi 0 q2 q2 1 q2 1 q2 1 1 1 q q q q q2 q2 1 uu ui uu 1 ui 1 1 uu ui uu ui uu 1 ui 1 8cu N 8ciN 8cm N 4 cu N 4 ciN 4 cm N 2 2 2 8cu N 8ciN 8cm N 1 1 1 1 1 1 1 1 1 0 0 1 1 00 1 ¨

8ciN 8cm N 4 ciN 4 cm N 2 8ciN 8cm N dter q2 q2 1 q2 1 q2 1 1 1 q q q q q2 q2 1 mu mi mu 1 mi 1 1 mu mi mu mi mu 1 mi 1 8cu N 8ciN 8cm N 4 cu N 4 ciN 4 cm N 2 2 2 8cu N 8ciN 8cm N q 1 q 1 1 1 q 1 1 q 1 0 ui 0 ui 1 1 uu ui 1 8ciN 8cm N 4 ciN 4 cm N 4 4 4 8ciN 8cm N q q q q 1 q q 1 q q 1 1 1 q q 1 q q q 1 q q 1 q q q q q 1 uu mu ui mi uu mu 1 ui mi 1 1 uu mi ui mu uu mu ui mi uu mu 1 ui mi 1 8cu N 8ciN 8cm N 4 cu N 4 ciN 4 cm N 4 4 4 8cu N 8ciN 8cm N q 1 q 1 1 1 q q q 1 q 1 0 mi 0 mi 1 1 mu mu mi 1 mi 1 8ciN 8cm N 4 ciN 4 cm N 4 4 4 4 8ciN 8cm N

000 0 0 0 0 0 0 1

Solid squares denote the focal gene copies and open squares other gene copies at the same locus. Females can be uninfected (open) or infected (shaded); coalescence (tenth state) can occur in either an uninfected or infected female or in a male. Ne With Sex-Ratio Distorters 315

Figure 3.—Scaling factor N e=2N for the effective population size of nuclear genes in an SRD-infected population, with xu ¼ xm. N e=2N is expressed as a function of the equilibrium frequency pˆ of the SRD and in relation to the following rates of maternal transmission xi=ðxi 1 xuÞ: 0.9 (solid line), 0.95 (dotted line), and 0.99 (dashed line). Note that pˆ can never be greater than any given transmission rate. The thick line igure = F 2.—Scaling factor N e 2N for the effective popula- gives the value for N e=2N that is obtained with the approxi- tion size of nuclear genes in an SRD-infected population, de- mation N e=2N ¼ cu 1 cm (see Equation 11). pending on the relative numbers xi of infected daughters and xu ¼ xm of uninfected daughters and sons that an infected female produces. The dashed red line marks the situation small, but the fraction ci of infected females in the where infected females produce as many offspring as unin- population takes a constant value. For given parameters fected females. xm ¼ xu, the parameter xi that is required to produce a certain equilibrium proportion ci of infected females in the population is given from Equation 3b by arises when there is perfect penetrance of the induced sex-ratio distortion, i.e., if all infected male offspring are 1 2ci killed or feminized and hence no infected males occur xi ¼ 1 xu: ð10Þ in the population. Perfect penetrance seems to be the 2 1 ci rule in both male-killing and feminizing Wolbachia and Inserting this into Equation 9 and assuming xm ¼ is therefore a very realistic assumption. With xm ¼ xu, the scaling factor by which the effective population size xu 0 yields is reduced is given by N eðxi; xuÞ 4 3 2 1 c ¼ c 1 c : ð11Þ N e 4xuð1024xi 288xi 16xi 2xi 1 1Þ i u m ¼ 3 2 2 : 2N 2N ð2xi 1 4xu 1Þ½ð512xi 144xi Þðxi 1 xuÞ 1 24xi 48xixu 1 10xi 4xu 1 ð9Þ Thus, the effective size of a population infected with a Figure 2 shows how this scaling factor depends on the ‘‘nearly neutral male killer’’ as defined above is given two parameters xi and xu (¼ xm). The strongest re- simply by twice the number of uninfected individuals in duction of Ne is obtained when xi is large and xm ¼ xu the population. This result has been anticipated are small. This corresponds to a high prevalence pˆ of the through verbal arguments (Sullivan and Jaenike SRD in the population and to a strongly female-biased 2006; Engelsta¨dter and Hurst 2007) and was also population sex ratio (low cm). Indeed, it can be seen confirmed in computer simulations (Engelsta¨dter urst from Equation 9 that limxu¼xm/0 N e ¼ 0 for any fixed and H 2007). 1 xi . 2 ; i.e., as the number of uninfected daughters and In Figure 3, the scaling factor for the effective males produced by infected females approaches zero, population size is shown in relation to the infection the effective population size converges to zero. frequency pˆ in the population (see Equation 2) and the We can also ask what the effective population size is maternal transmission rate of the endosymbionts, with a male-killing SRD that has a very high transmission xi=ðxi 1 xuÞ. This plot was obtained by a straightforward rate (xm ¼ xu 0), but that is selectively close to reparametrization in which xi and xu in Equation 9 are 1 ˆ neutral (xi 2 ). This latter assumption is probably very replaced by expressions involving p and the transmis- realistic for many male-killer systems in which the death sion rate. N e decreases with increasing prevalence of the of male offspring does only very weakly enhance the SRD and with increasing rate of maternal transmission. fitness of the surviving siblings. To obtain the effective The plot also indicates that for high rates of maternal population size for this case, we can derive an approx- transmission, Equation 11 provides a good approxima- imation of Equation 9 assuming that xm ¼ xu is very tion for N e. 316 Jan Engelsta¨dter

Special case 2: SRDs with perfect transmission but incomplete penetrance (xu ¼ 0,xm . 0): In this scenario, all offspring from infected females will also be infected, and a certain fraction of sons will be feminized or killed 1 by the bacteria. Provided that xi . 2 (either trough feminization or male killing with fitness compensation), Equation 2 shows that the SRD will spread to fixation in the population. Thus, there remain only infected females in the population that produce offspring with a male:female sex ratio of xm : xi. It is not surprising therefore that the scaling factor for the effective population size simplifies to

N e 4xixm ¼ 2 ¼ 4cicm ð12Þ 2N ðxi 1 xmÞ in this special case, which is Wright’s formula for the effective size of a population with arbitrary sex ratio (Wright 1931). Note that with feminizing SRDs, this Figure 4.—Scaling factor N e=2N for the effective popula- special case of xu ¼ 0 and xm . 0 may also ensue when tion size of nuclear genes in endosymbiont harboring popu- maternal transmission is not complete (see the lations that do not induce sex-ratio distortion (i.e., discussion). xu 1 xi ¼ xm). The dashed green line marks endosymbionts Special case 3: Endosymbionts that do not induce sex-ratio with a transmission rate xi=ðxi 1 xuÞ of 0.95. distortion (xu 1 xi ¼ xm): Although not the primary focus of this study, the model presented here also covers can be obtained. Although these parameters will depend maternally inherited endosymbionts that do not induce on the frequency of the endosymbionts, at equilibrium any sex-ratio distortion, but spread by enhancing the they are constant and therefore can be used to obtain N fitness of their hosts in some way. It is clear from the e as for other types endosymbionts. In most natural condition x . 1 that for this type of maternally in- i 2 systems where this has been studied, x will be very large herited endosymbiont to spread and be stably main- i and x will be rather small due maternal transmission tained in a population, x 1 x 1 x . 1 needs to be u u i m rates close to one and high offspring mortality in fulfilled. This means that infected females must have an incompatible crosses. Hence, only minor reductions in increased overall survival rate and/or fecundity com- N are expected with CI-inducing microbes. pared to uninfected females, as is the case for many e endosymbionts that provide their hosts with nutrients or protect them from parasites (e.g.,Aksoy 2003; Douglas DISCUSSION 2003; Oliver et al. 2003; Scarborough et al. 2005; Hosokawa et al. 2010). The coalescent effective size of populations infected No simple formula for Ne was obtained for this special with maternally inherited endosymbionts—in particular case. Figure 4 shows that for realistic parameter values those inducing a sex-ratio distortion in their hosts—was of xi and xu (with xm given by the sum of these two), the derived for both mitochondrial and nuclear host genes. reduction in Ne is only very weak. The strongest re- This was done using the method of the separation of duction is expected when xi is relatively small, but xu is time scales, a powerful tool that can be used to simplify large. This will be the case when maternal transmission various types of structured population, including geo- is very inefficient, but when the endosymbionts still graphical, age, or class structure (Mo¨hle1998; Nordborg produce a benefit for their hosts that is large enough to and Krone 2002; Ramachandran et al. 2008; see also ensure stable persistence in the population. Wakeley 2009, Chap. 6, for an accessible account of this Aside from endosymbionts that provide benefits (e.g., method). The model and parametrization used here was nutritional or protective) to their hosts, this special case kept deliberately simple in that the endosymbionts are also covers endosymbionts that induce cytoplasmic characterized by only three parameters (the relative incompatibility (CI), a reproductive incompatibility numbers of infected daughters, uninfected daughters, between uninfected females and infected males (re- and sons that infected females produce). Despite this viewed in Engelsta¨dter and Telschow 2009). Here, it simplicity, the model can be applied to a large variety of is essentially the offspring production of uninfected situations, including both male-killing and feminizing females that is altered by the endosymbionts, but microbes, arbitrary rates of maternal transmission, arbi- through rescaling, the parameters xu, xi, and xm relative trary penetrance of the sex-ratio distortion, and arbitrary to a total offspring production 1 in uninfected females reduction in overall fecundity of infected females. In Ne With Sex-Ratio Distorters 317 addition, the model also produces predictions on the be expected. The magnitude of these fluctuations will effective size of populations infected with maternally depend on the size of the population and the type and inherited symbionts that do not induce sex-ratio distor- penetrance of sex-ratio distortion (i.e., the strength of tion, but spread in the population by providing hosts selection for the SRD). For example, in many male- with some form of benefit or inducing cytoplasmic killing bacteria showing close to perfect maternal trans- incompatibility. mission but weak levels of fitness compensation for For mitochondrial genes, the effective population size females in a brood, we expect substantial fluctuations in is given simply by the number of infected females in the frequency around the equilibrium value. Such fluctua- population, independent of the type of sex-ratio distor- tions are likely to further reduce the effective population or other means of spread of the endosymbionts. tion size below the values predicted by this model, an This means that the strongest reduction in Ne is caused expectation stemming from previous results on pop- by endosymbionts that are stably maintained at a low ulations with changing size. [With fast fluctuations in prevalence in the population. Many male-killing bacte- population size, Ne is given by the harmonic mean of the ria are indeed present at a low prevalence (reviewed in actual sizes of the population, which is smaller than the Hurst et al. 2003), and endosymbionts where the arithmetic mean (Kimura and Crow 1963; Sjo¨din et al. induced phenotype is unknown are also often charac- 2005)]. For the case of male-killing bacteria, this terized by a low infection frequency (e.g.,Duron et al. reasoning is in accord with simulation results by 2008; Hilgenboecker et al. 2008). Because low preva- Engelsta¨dter and Hurst (2007). lence endosymbionts have a strong impact on mito- Another important assumption of the model pre- chondrial Ne and are common but easy to miss, these sented here is that the proportions of offspring that endosymbionts can easily confound interpretations of females produce depend only on their infection state. mtDNA polymorphism data (Hurst and Jiggins 2005). This assumption is violated in species with a ZW sex For nuclear genes, maternally inherited SRDs influ- chromosome system (i.e., where ZW individuals develop ence effective population size in the opposite direction: into females and ZZ individuals normally develop into the higher the SRD prevalence, the lower Ne. The males) that are infected with feminizing endosymbionts. general formula for nuclear Ne with endosymbiont This situation has been reported in the woodlouse infections is rather complicated, but can nevertheless Armadillidium vulgare as well as in the butterfly Eurema be calculated explicitly without computer simulations. hecabe, both infected with feminizing Wolbachia (Martin In some special cases, this formula can be reduced et al. 1973; Hiroki et al. 2002). There can be two types of considerably. In particular, the nuclear effective size of a females in such a population: ZW females (i.e., ‘‘real’’ male-killer infected population is often given to a good genetic females) and ZZ females (genetically male, but approximation by twice the number of uninfected feminized). Clearly, infected ZW females may produce individuals, a simple result obtained previously through all possible types of offspring if transmission is imper- verbal arguments and computer simulations (Sullivan fect, whereas ZZ females cannot produce uninfected and Jaenike 2006; Engelsta¨dter and Hurst 2007). In daughters. However, modeling has shown that the situations where feminizing microbes have become spread of the feminizing endosymbionts will eventually aylor fixed in the population, Ne is given by Wright’s well- lead to extinction of the W chromosome (T known formula for sex-biased populations (Wright 1990). Thus, at equilibrium all individuals will have a 1931). For endosymbionts that do not induce any ZZ constitution and their sex is determined solely by the sex-ratio distortion—including facultative mutualistic presence or absence of the endosymbionts, a condition endosymbionts or bacteria inducing cytoplasmic incom- observed—albeit complicated by an additional feminiz- patibility—Ne is also reduced, but only weakly. ing factor—in some populations of A. vulgare (reviewed Limitations of the model: Several simplifying as- in Rigaud et al. 1997). At equilibrium, this complicated sumptions have been made to obtain analytical expres- situation is therefore covered by the model presented sion of the effective population size. Most importantly, here, more precisely by special case 2, where xu ¼ 0 and right’s I assumed that the frequency of the endosymbionts in W (1931) formula for Ne in populations with the population is at a constant equilibrium. For mito- biased population sex ratio is valid. chondrial genes, it has been demonstrated that the Next, the method of separation of time scales used effective population size can also be strongly reduced here assumes that a within-population equilibrium of the during the spread of the SRD (Johnstone and Hurst distribution of gene copies is reached quickly relative to 1996). This is because the mitochondrial genotype that coalescent events. This assumption may be violated for happens to be associated with the ancestral bacterial mitochondrial genes when maternal transmission is very infection in the population will hitchhike along with the efficient. In this case, coalescence of two gene copies in SRD in the population, a process not covered by the uninfected females may occur faster than transition of current model. Moreover, even if the infection is at these copies into the class of infected females. In the equilibrium in the population, fluctuations around this extreme case of perfect maternal transmission—which equilibrium frequency due to random genetic drift can requires selective neutrality of the SRD if it is not to 318 Jan Engelsta¨dter become fixed in the population—no such transitions will was infected with male-killing Wolbachia at a very high occur, so that both classes can be regarded as isolated prevalence (99% infected females: Dyson and Hurst subpopulations. However, for large populations, infec- 2004), but within only 10 generations, a host suppressor tion frequencies that are not extremely small and with at gene effecting survival of infected males spread to least some low level of imperfect maternal transmission, fixation in this population, restoring the population the approximation through Mo¨hle’s (1998) theorem is sex ratio to 1:1 (Charlat et al. 2007). Consequently, this expected to hold. Furthermore, no such problems are population experienced a prolonged and drastic re- expected for nuclear genes, because here, there will duction in its effective size for nuclear genes (to necessarily be ample ‘‘gene flow’’ between infected and approximately 1% of the Ne without the infection), but uninfected individuals through males. then recovered quickly to its uninfected Ne. The re- Finally, I assumed that the effective size of the duced level of genetic variation that this reduction in Ne population in the absence of infection is given by the is likely to have caused is expected to be still strong in this census population size (i.e., N e ¼ N =2 for mitochondrial population. Moreover, the selective sweep that occurred and N e ¼ 2N for nuclear genes). In reality, many factors during the spread of the suppressor gene will have are known in addition to cytoplasmic SRDs that also lead caused further reduction in genetic variation at loci to reduced or increased effective population sizes. For linked to this suppressor gene. Without knowledge of example, if there is strong sexual selection in the the infection history of this population, the strong population so that males have a high variance in re- reduction in genetic variation might easily be mistaken productive success, this will decrease Ne (Kimura and for signs of demographic events like population bottle- Crow 1963). On the other hand, when bottlenecks necks. Thus, this example shows that cytoplasmic SRDs during mitochondrial transmission are large so that can severely confound interpretations not only of intra-individual mitochondrial variation (heteroplasmy) mitochondrial (Hurst and Jiggins 2005), but also of can build up, the effective population size for mitochon- nuclear DNA diversity. drial genes can also be larger than the number of females I thank Tanja Stadler, John Wakeley, Jon Wilkins, and an anonymous in the population (Birky et al. 1983). In many situations, reviewer for helpful comments on the manuscript and acknowledge modifications in Ne caused by cytoplasmic SRDs may support from the Swiss National Science Foundation. combine with other such modifications in a simple way (i.e., the factors quantifying these modifications in LITERATURE CITED Ne are multiplied), but it is not clear that this should always be so. 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Genetics 168: parasitic nature, host suppressor genes may often drive 1443–1455. them extinct, such footprints of past infections may be Dyson, E. A., and G. D. D. Hurst, 2004 Persistence of an extreme very widespread. A case in point is an Independent sex-ratio bias in a natural population. Proc. Natl. Acad. Sci. USA 101: 6520–6523. Samoan population of the butterfly Hypolimnas bolina. Dyson, E. A., M. K. Kamath and G. D. D. Hurst, 2002 Wolbachia For at least about 100 years until 2001, this population infection associated with all-female broods in Hypolimnas bolina Ne With Sex-Ratio Distorters 319

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APPENDIX

Here, the formulae for the expressions specifying the matrix P in Equation 8 as well as for Ne for nuclear genes in the general model are given. To simplify the notation, let us deﬁne the following placeholders:

A :¼ð2xi 1Þð4xi 1Þð4xi 1 1Þ;

B1 :¼ xm½2xið32xi 9Þ 1 1;

B2 :¼ xu½2xið32xi 1 9Þ1;

4 3 2 C1 :¼ 1024xi xu 224xi xu 88xi xu 1 22xixu xu;

4 4 3 3 2 2 C 2 :¼ 1024xi ðxi 1 xmÞ96xi 512xi xm 44xi 72xi xm 1 16xi 1 40xixm xi 2xm: The placeholders in matrix P in Equation 8 are then given by 320 Jan Engelsta¨dter

2 2 2 D1 :¼ 4xi xuð64xi 1 22xi 3Þ;

2 D2 :¼ð1 2xiÞ ½ð4xi 1Þxm 1 xu½ð4xi 1 1ÞB1 1 B2;

:¼ x x ½ x ð x Þ 1 x x ð x2 x Þ 1 x ; D3 8 i u A m 10 i 1 i u 144 i 6 i 11 u

:¼ ½ð x Þx 1 x ½ x ð x2 x 1 Þ 1 : Q 4 2 i 1 m u A m 64 i 18 i 1 C 1

The effective population size for nuclear genes in the most general scenario is given by

4xiQ N e ¼ 2N : ½ð2xi 1Þðxi 1 xmÞ 1 2xixu 1 xu½xuC 2 1 AB1ðxi 1 xmÞ 1 xuB2ð2xi 1Þ