Alternative evolutionary outcomes following endosymbiont‐mediated selection on male mating preference alleles

Abstract In many arthropods, intracellular bacteria, such as those of the genus Wolbachia, may spread through host populations as a result of cytoplasmic incompatibility (CI). Here, there is sterility or reduced fertility in crosses between infected males and uninfected females. As the bacterium is maternally inherited, the reduced fertility of uninfected females increases the frequency of the infection. If the transmission fidelity of the bacterium is less than 100%, the bacterium cannot invade from a low frequency, but if its frequency exceeds a threshold, it increases to a high, stable, equilibrium frequency. We explore the expected evolutionary dynamics of mutant alleles that cause their male bearers to avoid mating with uninfected females. For alleles which create this avoidance behaviour conditional upon the male being infected, there is a wide zone of parameter space that allows the preference allele to drive Wolbachia from the population when it would otherwise stably persist. There is also a wide zone of parameter space that allows a joint stable equilibrium for the Wolbachia and a polymorphism for the preference allele. When the male's avoidance of uninfected females is unconditional, the preference allele's effect on Wolbachia frequency is reduced, but there is a narrow range of values for the transmission rate and CI fertility that allow an unconditional preference allele to drive Wolbachia from the population, in a process driven by positive linkage disequilibrium between Wolbachia and the preference allele. The possibility of the evolution of preference could hamper attempts to manipulate wild populations through Wolbachia introductions.

uninfected females. Examples are male-killing and feminization (Engelstädter & Hurst, 2009). In male-killing, male offspring bearing the endosymbiont die, thus, in some circumstances, either relieving the competition experienced by their sisters or acting as a food source for these sisters. Another phenomenon is cytoplasmic incompatibility (CI), in which matings between infected males and uninfected females result in sterility or low fitness offspring, thus yielding, on average, a higher fertility for females with the endosymbiont, as was seen, for example, in the spread of the bacterium Wolbachia through Californian populations of Drosophila simulans (Turelli & Hoffmann, 1991). Early theoretical work (Caspari & Watson, 1959;Hoffmann, Hercus, & Dagher, 1998;Turelli & Hoffmann, 1991, 1995 demonstrated that the invasion of a population through a CI-generating bacterium would face difficulties if the fidelity of maternal transmission was less than 100% or if there is any fitness loss associated with infection. These factors, which tend to reduce infection frequency, are themselves frequency-independent. However, the advantage that infected females gain through CI increases with the frequency of the infection. This creates a situation where the absence of the bacterium is a stable equilibrium, but there is also potentially a high-frequency stable equilibrium for infection rate, which the population will move towards provided the initial frequency of infection exceeds an unstable threshold point. These complex dynamics have been of relevance to the use of Wolbachia in reducing insect-borne disease. It was demonstrated that Wolbachia, introduced into Aedes aegypti, reduced the ability of the mosquito to transmit dengue fever (Moreira et al., 2009). This result led to the manipulation of wild populations of the mosquito in Queensland, Australia, through the release of very large numbers of Ae. egypti infected with the wMel strain of Wolbachia, which blocks dengue transmission. The numbers had to be high since the Wolbachia was not only transmitted with less than 100% frequency, but also imposed a fitness cost on its bearers. Indeed, it was estimated that the unstable equilibrium that had to be exceeded was a Wolbachia frequency of around 30%, which was surpassed by the introductions, leading to near-fixation of the Wolbachia in these populations (Hoffmann et al., 2011). Subsequently, there has been evidence that Wolbachia can block transmission of Zika viruses in Ae. aegypti (Dutra et al., 2016) and, in some host species, some strains of Plasmodium (Moreira et al., 2009).
But the persistence of Wolbachia, with its harmful effects on host fitness, relies on the host failing to evolve to prevent the bacterium's effects. As with male-killing and feminization, with cytoplasmic incompatibility, there will be a selective advantage to alleles at nuclear (although not at mitochondrial or W chromosomal) loci that prevent the phenomenon. In addition to there being an advantage for alleles that prevent cytoplasmic incompatibility from occurring in crosses between infected males and uninfected females, it is clear that mutant alleles that will reduce the proportion of these CI-generating crosses will have an advantage. Champion de Crespigny, Butlin, and Wedell (2005) explored the expected outcomes in a model where a mutation causes females to avoid mating with infected males. The conclusions of this work were that, in the case where there was 100% fidelity in maternal transmission of the bacterium and no fitness costs associated with the infection, the infection would always spread and the mating preference would also spread. When there is less than 100% transmission fidelity, or when there is a fitness cost, the preference may prevent the infection's spread, given initial infection frequencies that would otherwise have permitted this, particularly when the initial frequency of the preference allele is high. The preference allele was always beneficial or neutral in the case where infection was either absent or at 100%. For this reason, there was no stable intermediate equilibrium for the preference allele. It moves to fixation or to a neutral intermediate equilibrium. This is because it was assumed that there was no male limitation, and females with a preference for uninfected males could always find these in a cost-free way.
Here, we examine a model where there is a preference allele expressed in males for infected females. However, our model for male preference is one that indirectly can impose a cost for the preference. And the male preference will, through its reduction in the proportion of CI matings that uninfected females undergo, reduce the frequency of the Wolbachia. (While the model is expressed in terms of Wolbachia, it is equally relevant to any other CI-inducing maternally inherited symbiont.) Males who choose infected females do so by reducing their matings with uninfected females by a proportion x relative to their proportions in random mating, and then, to this degree, compete with other males for matings with infected females. The consequence is that, since competition is now higher for access to the infected females, since these are chosen by males with the preference gene, males showing preference will have a reduced chance of mating overall. For this reason, given that the population contains both infected and uninfected females, the preference gene would be harmful when CI is not operating.
The preference shown by males for infected females could be either conditional (i.e. only shown by infected males) or unconditional (shown by all males). The advantage for the preference allele will be greater in the conditional case, but this requires the possibly biologically implausible assumption that the male's behaviour is conditional upon its own infection status.
When CI is complete, that is when all offspring of crosses between uninfected females and infected males die, a conditional preference can never be harmful, since the crosses that the preference gene prevents would all have been sterile. If, however, the sterility in CI crosses is not complete, the cost of competing for infected females could outweigh the reduced fitness of CI crosses for males exhibiting a preference. For an unconditional preference, non-CI-inducing uninfected with uninfected crosses will also be avoided, and the advantage of the preference will thus be reduced. This creates a subtle but important difference from earlier models (Champion de Crespigny et al., 2005) where preference is always neutral or beneficial.

| Modelling
The population dynamics of CI-inducing Wolbachia are complex. A simple analytical model predicts three equilibria for the infection frequency in the absence of mating preference but with less than 100% maternal transmission fidelity, which is shown in the Results section and Appendix 1.
In our model, we combine this CI model with the potential presence of an autosomal allele, M, that creates a preference in males for females that are infected. In the conditional model, the male preference only shows itself in males with the Wolbachia infection as well as the preference allele. In the unconditional model, the mate preference is shown by uninfected as well as infected males.
We thus assume that the single population consists of six genotypes and is panmictic except for any mating preferences shown by males. The genotypes are defined by U and I, denoting uninfected and infected, and MM, Mm and mm for the genotypes at the preference locus.

| For a conditional preference
Males with the IMM and IMm genotypes avoid matings with uninfected females, with avoidance of x and dx, respectively, and so the proportion of the males competing for matings with uninfected females is 1 − x(p IMM + dp IMm ), which we represent by C U .
Thus, for an uninfected female genotype i, of frequency p Ui , the relative probabilities of mating with different genotypes of males are as follows: The avoidance, of strength x and dx, respectively, by infected MM and Mm males, of uninfected females, will release MM and Mm males to compete for the infected females. The impact on competition for the infected females of these extra infected males released will be proportional to the relative proportions of uninfected and infected females, represented by p U /p I . The competition for infected females, which we call C I , is thus 1 + (p IMM + dp IMm )xp U /p I . Thus, for an infected female genotype i, the relative probabilities of mating with different genotypes of males are as follows: For infected MM males:  To test the impact of a finite population size on this model, the program was modified to include a multinomial sampling of genotypes in a finite population of size N. The proportions of the six genotypes above are calculated analytically, and then, a population for the next generation is created by multinomially sampling these six genotypes N times, with replacement. Then, the numbers in the sample are converted to frequencies that are used for the next generation.

| For an unconditional preference
Now all males, whether or not they are infected, avoid mating with the uninfected females. So the competition for uninfected females, C U , is 1 − x(p IMM + dp IMm + p UMM + dp UMm ). Thus, for an uninfected female genotype i, of frequency p Ui , the relative probabilities of mating with different males are as follows: The avoidance, of strength x or dx, by all MM and Mm males (whether infected or not), of uninfected females, will release MM and Mm males to compete for the infected females. The competition for infected females, or C I , is thus 1 + (p IMM + dp IMm + p UMM + dp UMm )xp U /p I . Thus, for an infected female genotype I, the relative probabilities of mating with different genotypes of males are as follows: For infected MM males: The model is expressed as a C++ program (see Data S1), into which is input: The transmission rate, c, of the Wolbachia from mothers to offspring; The fertility, f, of crosses between infected males and uninfected females; The initial frequency of the Wolbachia infection, W; The initial frequency of the preference mutation, M; The strength of the effects of the preference mutation, x; The dominance, d, of M, where 1 is fully dominant, and 0 is fully recessive; and The number of generations of simulation.
From these inputs, the program creates the initial distribution of p UMM , p UMm , p Umm , p IMM , p IMm and p Imm by assuming Hardy-Weinberg equilibrium and linkage equilibrium (although the population that evolves does not show these properties).

| RE SULTS
The results presented here include an overview of the established theory of the dynamics of endosymbionts creating cytoplasmic incompatibility. This is followed by the results of simulations of the outcome of the conditional model, and a demonstration of the conditions under which a preference allele could either eliminate or come into a stable equilibrium with a Wolbachia infection.

| Cytoplasmic incompatibility
The fundamental model of CI has been explored by previous authors (Caspari & Watson, 1959;Turelli & Hoffmann, 1995). Our simplified model includes random mating, but the absence of any cost or benefit from the bacterium other than from CI. p is the proportion of surviving offspring that come from mothers with Wolbachia (called I as opposed to U). Appendix 1 shows that this system has three equilibrium points, a stable equilibrium at p = 0, a high stable equilibrium p, and an intermediate unstable equilibrium p.
An example of these equilibria is shown in Figure 1, based on the model's simulation when the preference allele is absent.

| Simulation results: conditional model
In the conditional model, a mutation, M, arises that causes males that are I to avoid any mating with females that are U. We ask whether such a mutation can spread and its impact on the frequency of Wolbachia. In each of two sets of conditions that have been considered (i.e. c = 0.9, f = 0.5; c = 0.8, f = 0), the outcome observed is that a dominant preference allele (d = 1) with full penetrance (x = 1) will spread through the population and cause the elimination of the Wolbachia from the population. This is because the allele is preventing I males from mating with U females, causing them to compete for the increasingly scarce I females.

F I G U R E 1
The situations when the transmission rate, c = 0.9, and the fertility in cytoplasmically incompatible crosses, f = 0.5. There are equilibrium points at 0 (stable), 0.307 (unstable) and 0.804 (stable), which are revealed by changes in the frequency of the Wolbachia with time, given different starting frequencies.
Thus, for a given set of parameters of the Wolbachia infection, the initial frequency will determine whether the population moves to the high stable equilibrium point or whether the bacterium will be lost

Time in generaƟons
As I females become rarer, the lowered probability of males obtaining a mating starts to outweigh the fitness cost that comes from CI (since, in this case, the CI crosses still have a fertility that is half that of the other crosses). The preference allele also now spends more time in uninfected males, and its effects are thereby diminished. In Figure 2b, where the CI crosses are completely sterile, competing for the scarce I females can never be worse than mating with U females, and the preference allele can never be disadvantageous. In both cases, once Wolbachia has gone, the preference allele is in a neutral equilibrium, since this allele expresses a phenotype only in infected males.

F I G U R E 2
The impact of introduction of a dominant preference allele M (of maximum strength: x = 1) at a frequency of 0.002 into a population with a Wolbachia infection initially at a frequency of 0.6. In (a), the parameters are c = 0.9, f = 0.5, and in (b), they are c = 0.8, f = 0. In each figure, the uppermost (green) line shows the movement of the Wolbachia frequency to a stable equilibrium in the absence of the preference allele. The red line, showing very similar initial frequency changes, followed by a decline, is the Wolbachia frequency when the preference allele is introduced. The lowest (blue) line is the frequency of the preference allele If the value for c is raised, with f still equals to 0.5, there can be a joint stable equilibrium generated, and example of which is shown in Figure 3. Figure 3 gives an example where both the Wolbachia and the preference allele reach a joint stable equilibrium. Here, the preference allele is introduced at a low frequency, but, as we have seen that there are three equilibria that exist for the Wolbachia without the preference allele, a stable equilibrium at zero, an unstable equilibrium and a high stable equilibrium, we can ask whether the initial frequency of the preference allele influences how high the initial frequency of the Wolbachia has to be in order not to be lost.  Losses of Wolbachia can sometimes be triggered by the frequency of M drifting to considerably higher than its equilibrium frequency, which can be followed by a rapid decline and loss of the Wolbachia. Figure 5a shows, at around generations 150-170, a decline in the Wolbachia frequency following a high frequency of M being reached, although in this case the Wolbachia recover. If Wolbachia is lost, M becomes neutral and rapidly drifts to fixation or loss. As can be seen in Figure 5a, M fluctuates greatly when population size is small, and is usually seen below its stable equilibrium point. If it is lost, the probability of subsequent loss of Wolbachia is greatly diminished. Figure

| Simulation results: Unconditional model
Now males with the preference allele will avoid mating with females that are uninfected, whether or not the males are themselves infected. In these circumstances, the advantage for the preference allele will be less, as some full fertility crosses as well as CI crosses are being avoided by the males showing the preference.

| Approximate analytical results
We have looked analytically at a model where we, inaccurately, assume that there is linkage equilibrium between the presence of If a population has had its Wolbachia infection eliminated by a conditional preference allele, the preference allele may persist and immunize the population, to some restricted degree (see Figure 4) This would result in a population where infected males showed a consistent partial avoidance of mating with uninfected females. The other situation would be the joint stable equilibrium where infected male avoidance of uninfected females could be complete or incomplete, but will be shown only by a subset of the males.
The model that has been outlined here has assumed that there is no advantage to the Wolbachia infection and that its spread is solely through CI. Clearly, if there was a substantial fitness gain associated with Wolbachia, then a preference allele that reduced or prevented CI would spread due to its selective advantage, but would not be able to eliminate the Wolbachia (provided that the Wolbachia selec-  (Vala, Egas, Breeuwer, & Sabelis, 2004).
While an unconditional preference mutation might seem easier to achieve than a conditional mutation, it is possible that if an unconditional mutation were to spread to the joint stable equilibrium, further mutational changes that made the preference conditional upon the male infection status could spread and eliminate the Wolbachia.
Evidence that Wolbachia can enhance fitness is inconsistent (Fry, Palmer, & Rand, 2004;Fry & Rand, 2002;Harcombe & Hoffmann, 2004;Ming, Shen, Cheng, Liu, & Feng, 2015). It has been argued that, since selection for alleles that prevent the effects of CI will only be strong when CI is frequent, which requires an intermediate value for the infection frequency (Sahoo, 2016), there will be few examples of evolution of host countermeasures to CI, since populations will typically be at their stable equilibria of either very high or zero infection frequencies. But our models suggest that selection for preference alleles could be strong, if mutation could produce the required alleles.
As the standard model for a Wolbachia that is unstable in transmission but spread by CI suggests that loss of the endosymbiont is stable, it is not clear how Wolbachia can ever invade, unless it conveys a direct fitness advantage in females. It could be through genetic drift. If c is very close to 1.00, and f is 0, the unstable equilibrium predicted in the absence of preference and selection is a Wolbachia frequency of approximately 1 − c, which might be attained by drift if it is just one or two per cent. An estimate of c in wild populations of D. melanogaster is 0.974 (Hoffmann et al., 1998). The population genomics of Wolbachia in this host shows congruence with mitochondrial DNA variants, indicating a single infection, although one that (Richardson et al., 2012) is subsequently affected by losses of Wolbachia, with c < 1.00.
We have thus seen that Wolbachia can potentially be eliminated from populations through the evolution of a preference allele in males that causes the avoidance of cytoplasmically incompatible crosses (just as a preference allele acting in females could also have this effect (Champion de Crespigny et al., 2005) infection statuses may make the mutation rate to alleles with this property restrictively low.
As with any population genetics model which includes evolution towards an equilibrium state, it is uncertain whether the values of the dynamic parameters will remain constant in time for long enough for the equilibrium value defined by these parameters to be reached. The spread of Wolbachia through CI-driven selection will, in most cases, be faster than its loss through incomplete transmission.
However, this is a situation where the processes of sterility, preference, and incomplete transmission are intrinsic to the biology of the two interacting species, rather than being environment-dependent. Their parameters might thus be less labile than those in models where environments have key effects.

ACK N OWLED G M ENTS
We thank two anonymous reviewers for constructive comments that have greatly improved the manuscript.
infected females and a proportion 1 − pc of crosses involve uninfected females, and, as the M + fitnesses are all a product of r and the mm are all a product of 1 − r, it is possible to calculate the fitnesses of M + and mm males.
When these fitnesses are equal, This is true if pcx(1 − pc) = 0, that is if there is no variation in the population (pc(1 − pc) = 0), or if x = 0 (the preference allele has no phenotype). Alternatively, it is true if The numerator thus, for equal fitness, must equal 0, and This is thus the predicted equilibrium value of r.
Equally, for a given r value, we can predict the equilibrium p. This will come when the fitness of the population is equal to c.
It is U females that show a loss in fitness. The males that will mate with U females will include: (If x < 1.0) I males with the preference mutation, which will constitute pcr(1 − x) I males without the preference mutation, which will constitute pc(1 − r) U males, which will constitute 1 − pc.
As the first two types of crosses will be CI crosses, with fertility f, the overall fertility (fitness) of U females will be which can be simplified to This is the fitness of U females, which constitute a proportion 1 − pc of the female population. The pc I females have a fitness of one. Thus, the population fitness is At equilibrium p, this must be equal to c.
There are two solutions for p, but the upper, stable, equilibrium is given by which is the same as (3) when r = 0 (i.e. when there is no preference allele) We look at our simple equilibrium when M is dominant, x = 1, f = 0.5 and c = 0.95. This moves in the simulation to a joint stable equilibrium where p is 0.89848, and r is 0.60079. The formula (4) predicts that, for p = .89848, equilibrium r should be 0.53950, and formula (5) predicts that, for r = 0.60079, equilibrium p should be 0.90403. Trial and error reveal that, using (4) and (5), the predicted joint stable equilibrium is p = 0.91258 and r = 0.53564. The positive linkage disequilibrium between the Wolbachia and the preference allele, which is not included in these approximate results, has the effect that, in the simulations, the preference allele achieves a higher fitness and thus frequency, since it is found preferentially in infected males, where its advantage is greater, and this linkage disequilibrium also has the effect that the preference allele is more effective in reducing the frequency of the Wolbachia.
Note that, when f = 0, (4) predicts that there can be no equilibrium r value that is one or less. r can only have a stable equilibrium value if f (the CI fertility) is greater than zero.

Unconditional model
We assume that there is linkage equilibrium between the infection and M allele.
We can consider the matings with uninfected females.
Of males available, the proportion with M alleles is r, and the proportion infected is pc.
All males showing preference lower their matings with uninfected females by a proportion x. This means that the relative proportions of males of different genotypes mating with uninfected females are as follows: Alternatively, it is true if So, The numerator thus, for equal fitness, must equal 0, and This is thus the predicted equilibrium value of r.
Simulations of an unconditional dominant preference allele with x = 1, f = 0.5 and c = 0.90 gives a joint stable equilibrium where p is 0.785056 and r is 0.422125 (summing homozygotes and heterozygotes for M). The simulated value of r is very much greater than the value of 0.27848 predicted by (6).
What is the equilibrium p? This will come when the fitness of the population is equal to c.
It is U females that show a loss in fitness. The males that will mate with U females will include: (If x < 1.0) I males with the preference mutation, which will constitute pcr(1 − x) I males without the preference mutation, which will constitute pc(1 − r) (If x < 1.0) U males with the preference mutation, which will constitute (1 − pc)r(1 − x) U males without the preference mutation, which will constitute (1 − pc)(1 − r) As the first two types of crosses will be CI crosses, with fertility f, the overall fertility (fitness) of U females will be which can be simplified to or pcf + 1 − pc.
This is the fitness of U females, which constitute a proportion 1 − pc of the female population. The pc I females have a fitness of one. Thus, the population fitness is