Curtis M. Lively, Department of Biology, Indiana University, Bloomington, IN 47405, USA. Tel.: +1 812 855 1842; fax: +1 812 855 6705; e-mail: email@example.com
The Red Queen hypothesis posits a promising way to explain the widespread existence of sexual reproduction despite the cost of producing males. The essence of the hypothesis is that coevolutionary interactions between hosts and parasites select for the genetic diversification of offspring via cross-fertilization. Here, I relax a common assumption of many Red Queen models that each host is exposed to one parasite. Instead, I assume that the number of propagules encountered by each host depends on the number of infected hosts in the previous generation, which leads to additional complexities. The results suggest that epidemiological feedbacks, combined with frequency-dependent selection, could lead to the long-term persistence of sex under biologically reasonable conditions.
The Red Queen hypothesis has nonetheless been controversial since its inception. In a groundbreaking paper, W.D. Hamilton showed using a simulation model that sex could be favoured by host–parasite coevolution (Hamilton, 1980). However, as pointed out by May and Anderson, Hamilton’s model, which incorporated strong frequency-dependent selection, did not rely on parasites as the unique mechanism for the source of selection (May & Anderson, 1983). Specifically, Hamilton’s model did not incorporate any of the epidemiological details associated with parasites, such as a threshold density in susceptible host population and time-lagged responses by the parasites. In May and Anderson’s model, where such details were incorporated, the conditions under which host–parasite coevolution could select for sex required very high levels of parasite virulence, leading them to suggest that the parasite hypothesis had limited application to real-world situations (May & Anderson, 1983).
May and Anderson also pointed out, however, that the epidemiological details in their model were not ‘brought down from Mount Sinai,’ and that other assumptions might revive the theory. Since that time, however, very little theoretical effort has gone into expanding on the epidemiological details of infection. A notable exception comes from work by Galvani et al. (2001, 2003). Using epidemiological models, they showed that sex could be maintained in parasitic helminthes when interacting with the vertebrate immune system. In the present study, I consider some different epidemiological features of host–parasite interactions. Most importantly, I relax the assumption of many models that the number of parasite exposures per host is constant overtime (usually equal to one). Here, the mean number of propagules encountered by each host depends on the number of hosts infected in the previous generation, and, as such, it can change over time. The results suggest that host sex can persist over a broad range of parasite fecundities.
Most models of the Red Queen hypothesis assume that the probability of infection for a particular host genotype depends only on the frequency of the matching parasite genotype in the parasite population. This, in turn, assumes that every host is exposed to exactly one parasite. As such, most models do not allow for numerical dynamics in the parasite population, which could change the probability of exposure or the number of exposures over time. For example, it is possible that when a clone spreads into a sexual population, the total number of infections would increase, thereby increasing the number of exposures to parasites. In the simulations that follow, I kept track of the number of infected individuals in each generation, and the number of infected hosts in one generation directly affected the number of exposures to parasites in the next host generation.
The basic simulation is similar in structure to a previous model (Lively, 2009). I assume a haploid host with two loci, each with three alleles, which determine susceptibility/resistance to parasites. Parasites were also treated as having three alleles at each of two loci, and they were assumed to be asexual. Parasites that matched the host at both loci infected the host; parasites that did not match the host were killed by the host’s innate immune system, resulting in strong selection on the parasite, which is a reasonable assumption for obligate parasites (Salathe et al., 2008). This kind of genetic interface for infection genetics is known as the matching alleles model, and it serves as a framework for modelling the self–nonself recognition in invertebrates (Frank, 1993; Otto & Michalakis, 1998). The matching alleles model can be seen as being at one end of a continuum between ‘gene-for-gene’ models in plant pathology and matching allele models (Parker, 1994; Agrawal & Lively, 2002). Elsewhere, we have found that, over most of the continuum, the genetical dynamics are similar to the pure matching allele models (Agrawal & Lively, 2002).
In an annual host population, the number of sexual individuals having the ijth genotype at time t +1 (), assuming no recombination between resistance loci, is:
where s gives the frequency of males in the sexual population and Pij gives the probability of infection for the ijth host genotype. The variable WI gives the fitness of infected hosts, whereas WU gives the fitness of uninfected hosts. In the presence of recombination between resistance loci, the variable was adjusted to account for the effect of recombination between the two resistance loci in the sexual host population (see Hartl & Clark, 1989). Specifically, the number of sexual individuals of the ijth type after recombination () is:
where N′ is the total host population size at time t +1;ρ is the frequency of recombination; qij is the frequency of the ijth genotype in the host population after selection; qi is the frequency of the ith allele at locus one after selection; and qj is the frequency of the jth allele at locus two after selection [eqn 2 above is also the corrected form of equation (10) in Lively (2009)].
Similarly, the number of asexual individuals having the ijth genotype at time t +1 is:
where Aij is the number of infected asexual clones having the ijth genotype at time t. Hence, the only differences between the recursions for sexual and asexual genotypes are that the sexuals pay the cost of producing males, and they have recombination between the resistance loci (The results presented below were remarkably robust to the rate of recombination as well as to the assumption that hosts and parasites were haploid).
The probability of infection for either a sexual or asexual individual having genotype ij at time t +1 is given as:
where β is equal to the number of propagules produced by each infection that also make contact with a host; thus, β can be thought of as the realized fecundity of the parasite. On the right-hand side of eqn 4, β [SijI + AijI]/N′ gives the Poisson mean number of exposures for hosts having genotype ij to parasites having a matching (infective) genotype. The exponential term (exp (−β [SijI + AijI]/N′) gives the probability of the zero class, which here is the probability of not encountering a matching parasite. Equation 4 thus gives the probability of encountering one or more matching parasites. Hence, this formulation assumes that one exposure is sufficient for infection. I also assume that if there are two or more exposures to matching parasites, then only the first parasite establishes, meaning that there are no coinfections in the model. This latter assumption is conservative with respect to virulence whenever unrelated coinfections lead to an increase in virulence (van Baalen & Sabelis, 1995; May & Nowak, 1995; Frank, 1996).
The number of secondary infections, R0, produced following the introduction of a single infected individual having genotype ij into a large host population is , where gij is the frequency of the ijth genotype in the host population (Lively, 2010c). Thus, the spread of infection by an asexual parasite strain that can infect the ijth host genotype requires that . The mean value for R0, averaged over all parasite genotypes, is simply , where G gives the number of genotypes in the population (Lively, 2010c). The spread of the disease requires that R0 >1, which in turn requires β > G if all host genotypes are equally common.
Virulence (V) was calculated in the simulation runs as
where WU gives the fitness of uninfected host individuals and WI gives the fitness of infected individuals. In the present simulations, the fitnesses for the annual host were further defined (following Maynard Smith & Slatkin, 1973) as:
where bU gives the number of offspring produced by uninfected hosts in the absence of competitors and bI gives the number of offspring produced by infected hosts in the absence of competitors. Similarly, aU scales the effect of total host density on offspring production in uninfected hosts, and aI scales the effect of total host density on offspring production by infected hosts. Defined in this way, virulence potentially has both density-dependent and density-independent aspects, and it depends on the relative fitness of infected individuals (Lively, 2006). This formulation has two advantages that were not possible in the linear formulations that I used previously (Lively, 2009). First, the density-dependent effects are easily separated from the density-independent effects. For example, for aI = aU, virulence becomes simply 1 − (bI/bU) for all host densities. Similarly, for bI = bU, virulence depends only on host density. Second, for x =1, the population dynamics are stable for all values of b (see Doebeli & de Jong, 1999). In the present simulations, I set aI = aU, so virulence was constant over all densities, and x =1, so the numerical dynamics were inherently stable in the absence of infection.
The deterministic simulations all began with a ‘burn-in’ period of 1000 generations, during which the presence or absence of Red Queen dynamics in the sexual population was observed. (The oscillation of host/parasite genotype frequencies over time is known as ‘Red Queen dynamics.’) After the burn-in period, a single clonal individual (hence a single genotype) was introduced into the sexual population, and the simulation was run for another 1100 generations. The goal was to determine the effect of β on the genotype frequency changes in the sexual population during the burn-in period and on the fate of the clone. I assumed that males constituted half of the sexual population (s =0.5) and that the probability of recombination between loci in the sexual population was 0.20. The other parameter values used were as follows: bU = either 10 or 20; bI = 3; and aU = aI = 0.001. Thus virulence, V, was equal to 0.7 for bU = 10 (i.e. V =1−(3/10) = 0.7), and virulence was equal to 0.85 for bU = 20 (i.e. V =1−(3/20) = 0.85). Finally, I allowed migration into the host population. Infected migrants entered the host population stochastically with a probability of 0.02 for each genotype, and uninfected migrants entered the host population with a probability of 0.10 for each genotype.
The results broke nicely into several zones that were defined by whether the parasite became endemic in the host population and whether Red Queen dynamics were observed.
Zone 1: β ≤ 1
As expected, for values of β that were less than or equal to one, the parasite did not spread into either a sexual or an asexual host population. Thus, the clone invaded and went to near fixation (Fig. 1). The number of sexual individuals that could be maintained at migration–selection balance in the absence of parasites was about one (Fig. 1). In the absence of migration, the sexual population was completely eliminated (results not shown).
Zone 2: G > β > 1
When β was greater than one, but less than the number of genotypes in the sexual population, the parasites still did not spread in the sexual population (R0 less than one). Thus, the sexual population was not infected, and it was not undergoing Red Queen dynamics when the clone was introduced in the population (Fig. 2). Under these conditions, the clone invaded and went almost to fixation, with a very small sexual population being maintained (Fig. 3a,d). However, as the clone became common, the number of genotypes in the host population was effectively reduced (G→ 1), which allowed the parasite to spread in the host population (β > G). Following the initial spread of the parasite, the number of parasite exposures per host jumped dramatically (Fig. 3b,e), and the frequency of infection in the asexual subpopulation jumped from near zero to over 80% (Fig. 3c,f). After the asexual population became heavily infected, the number of sexual individuals increased in the population (Fig. 3a,d), and the frequency of infection decreased (Fig. 3c,f). The end result was the stable coexistence of sexual and asexual individuals at the point where the cost of producing males was balanced by the advantage of parasite evasion in the sexual population (see Lively, 2010b).
This set of results illustrates an important point: the lack of infection in sexual populations cannot be taken as evidence that parasites would not be important for the local persistence of sex in the long term. In a real sense, the invasion of the clone changed the local ecology; by spreading into the sexual population, it set up the conditions for the spread of infection in an otherwise disease-free population. This then allowed the sexuals to spread when rare, as they were mostly free of infection.
Zone 3: β slightly greater than G
The third zone is more familiar to students of the Red Queen. In this zone, β is slightly greater than the number of sexual genotypes, so the parasite could spread, and it became endemic in the sexual population (Fig. 4c,f). The sexual genotypes oscillated in the standard fashion known as Red Queen dynamics, indicating that the different sexual genotypes oscillate in turn (Fig. 2). Following invasion by the clone, the number of parasite exposures spiked (Fig. 4b,e), and the frequency of infection in the host population increased dramatically (Fig. 4c,f). The clone was then driven down in frequency, and the population became a stable mixture of sexual and asexual individuals, which was numerically dominated by the sexuals (Fig. 4a,d). Surprisingly, however, zone 3 can be very narrow (Fig. 2). This result shows that Red Queen dynamics are not necessary for the parasite theory of sex.
Zone 4: β much greater than G
In zone four, the number of successful propagules is more than slightly greater than the number of genotypes in the sexual population. In this zone, the parasite-mediated oscillations in genotype frequencies did not occur prior to invasion of the clone (Fig. 2). The reason for this is that hosts were receiving multiple exposures and most of the population was infected (Fig. 5a,d). Hence, there was little to no variation for fitness among the sexual host genotypes. However, when the clone invaded the sexual population, the parasites quickly evolved to specialize on the clone’s genotype, and the frequency of infection in the sexual population decreased (Fig. 5c,f). Here, the number of exposures per host was not dramatically changed (as above) following the invasion of the clone (Fig. 5b,e), but the sexual population still persisted in the population, where the frequency of sexuals depended on virulence (compare Fig. 5a and d).
The model presented here differs from most previous models in several ways. The most important difference is that the number of exposures to parasites was not constant but rather depended on the number of infected hosts in the previous generation. Hence, the probability of infection relied on more than just the frequency of matching parasite genotypes, which implicitly assumes that every host is exposed to exactly one parasite. In addition, the present model allowed for variation in the number of exposures. The mean number of exposures is simply the average number of parasite propagules per host. The probability of exposure is calculated as one minus the probability of the zero class in a Poisson distribution (), where the mean (λ) is number of parasites per host. Hence, the possibility exits that some hosts are not exposed to parasites, unless the mean number of parasites per host is high. For example, if the mean number of parasites per host is 1, then about 37% of the hosts are not exposed to parasites; if the mean number of parasites per host is 4, then only 2% of hosts are not exposed to at least one parasite. Thus, small changes λ in can lead to big changes in the strength of parasite-mediated selection.
The results of this study show that host–parasite coevolution can lead to the stable coexistence of sexual and asexual (mixed) populations over a wide range of values for realized parasite fecundity, β (Fig. 1). They also show that oscillations in genotype frequencies over time (Red Queen dynamics) are not required for parasites to select for sex (see also Howard & Lively, 2003). In fact, the results suggest that the parameter space for which sexual genotypes are oscillating prior to the introduction of the clone can be quite narrow (Figs 1 and 2). As such, it would appear that Red Queen dynamics are neither necessary nor sufficient for the Red Queen hypothesis for persistence of sex. On the other hand, genetic drift in small populations might be sufficient to perturb an otherwise stable polymorphism to show oscillatory behaviour (Kouyos et al., 2007).
The results also show that parasite-free sexual populations cannot be taken as evidence to suggest that parasites could not become an important source of selection. The reason is that the spread of a clone into a sexual population reduces the genetic diversity of the host population, thereby increasing R0 for the matching parasite genotype. If R0 for the matching parasite increases to be greater than one, then the infection will spread in what was a previously disease-free population (Fig. 3). In addition, the number of parasite exposures per host can increase dramatically as the infection spreads (Fig. 3), which increases the overall strength of selection against the clone. This combination of epidemiological feedbacks, along with the coevolutionary response of the parasite, can prevent fixation of the clone in the short term, allowing for the persistence of sexually reproducing females.
In this study, a single clone was placed in competition with a sexual population. Elsewhere, we have found that sequentially introducing different clones into the sexual population would lead to replacement of the sexual population by a genetically diverse clonal population, unless the reproductive advantage of clones was blunted over time by Muller’s ratchet (Howard & Lively, 1994; Lively & Howard, 1994). Similarly, models that initiate the clonal population as ‘fully saturated’ (meaning that the clonal population contained all the genotypes contained in the sexual population) tend to be less favourable to the parasite theory of sex (e.g. May & Anderson, 1983; Otto & Nuismer, 2004), presumably because it is difficult for the sexual population to move away from the clonal population in genotypic space (review in Lively, 2010a). For similar reasons, parasite-mediated selection may not favour sexual reproduction in species for which individuals could potentially produce both sexual and asexual offspring within the same brood, unless the parasites are transferred from the mother to her offspring (Agrawal, 2006). Finally, parasite-mediated selection is unlikely to prevent the invasion of alleles for self-fertilization, but it can prevent the fixation of such alleles, leading to stable mixed mating in the host population (Agrawal & Lively, 2001). In this study, I found that an obligately sexual population could coexist with an obligately asexual population, provided the asexual population consisted of a single resistance genotype.
The results presented here differ from those found in the epidemiological model by May & Anderson (1983), but the assumptions also differ. May and Anderson assumed that infected individuals either recovered or died. They found that host–parasite coevolution would only favour sexual reproduction if none of the infected hosts recovered, so virulence was extremely high. In the present model, I assumed that infection by parasites did not increase the mortality rate but rather reduced the birth rate of infected individuals. Under these assumptions, parasites selected for sexual reproduction when virulence was greater than 1/(2f − g), where f is the frequency of infection in the asexual population and g is the frequency of infection in the sexual population (see Lively, 2010b). It is possible that the frequency of infection in the asexual population was not high enough in May and Anderson’s model to select for sexual reproduction. May & Anderson (1983) also assumed that the clonal population started with the same genetic diversity as in the sexual population (i.e. ‘fully saturated’), which (as indicated above) reduces the advantage of sex. In any case, May & Anderson (1983) were correct to point out that different epidemiological assumptions might be more favourable to the Red Queen.
I thank Lynda Delph, Dieter Ebert, Kayla King, Maurine Neiman and two anonymous reviewers for helpful comments on the manuscript. This work was supported by the US National Science Foundation (DEB-0640639).