On the mod resc Model and the Evolution of Wolbachia Compatibility Types

Corresponding author: Sylvain Charlat, Institut Jacques Monod, Laboratoire Dynamique du Génome et Evolution, CNRS, Universités Paris 6 & 7, 2 Pl. Jussieu, 75251 Paris Cedex 05, France., charlat{at}ijm.jussieu.fr (E-mail)

Communicating editor: M. A. F. NOOR

	ABSTRACT

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Cytoplasmic incompatibility (CI) is induced by the endocellular bacterium Wolbachia. It results in an embryonic mortality occurring when infected males mate with uninfected females. The mechanism involved is currently unknown, but the mod resc model allows interpretation of all observations made so far. It postulates the existence of two bacterial functions: modification (mod) and rescue (resc). The mod function acts in the males' germline, before Wolbachia are shed from maturing sperm. If sperm is affected by mod, zygote development will fail unless resc is expressed in the egg. Interestingly, CI is also observed in crosses between infected males and infected females when the two partners bear different Wolbachia strains, demonstrating that mod and resc interact in a specific manner: Two Wolbachia strains are compatible with each other only if they harbor the same compatibility type. Here we focus on the evolutionary process involved in the emergence of new compatibility types from ancestral ones. We argue that new compatibility types are likely to evolve under a wider range of conditions than previously thought, through a two-step process. First, new mod variants can arise by mutation and spread by drift. This is possible because mod is expressed in males and Wolbachia is transmitted by females. Second, once such a mod variant achieves a certain frequency, it can create the conditions for the deterministic invasion of a new resc variant, allowing the invasion of a new mod resc pair. Furthermore, we show that a stable polymorphism might be maintained in natural populations, allowing the long-term existence of "suicidal" Wolbachia strains.

CYTOPLASMIC incompatibility (CI; reviewed in HOFFMANN and TURELLI 1997 ; CHARLAT et al. 2001 ) is induced by the maternally inherited endocellular bacterium Wolbachia, widespread in Arthropods (WERREN et al. 1995 ; JEYAPRAKASH and HOY 2000 ). This phenomenon results in a more or less intense host embryonic mortality, occurring when infected males mate with uninfected females, while the three other types of crosses are fully fertile (unidirectional incompatibility, Fig 1A). As a consequence of unidirectional incompatibility, infected females are normally fertile when mating with both infected and uninfected males, while uninfected females suffer a fertility deficit when mating with infected males. The more frequent the infected males, the more frequent are the crosses detrimental to uninfected females. Because Wolbachia is transmitted by females only, infected cytoplasms are selected for in a positively frequency-dependent manner, allowing the bacterium to spread through the population and then maintain itself. Considering the invasion dynamics in more detail, theoretical analysis (CASPARI and WATSON 1959 ; FINE 1978 ; HOFFMANN et al. 1990 ), together with empirical data (TURELLI and HOFFMANN 1995 ), highlighted the importance of three main parameters: (i) CI level (the percentage of embryos killed by CI in incompatible crosses); (ii) the fitness effect of infection on hosts (apart from CI); and (iii) the bacterial transmission efficiency from mothers to offspring. The above studies showed that the frequency of infected individuals presents a stable equilibrium depending on these three parameters. This stable equilibrium frequency is 1 if maternal transmission is perfect and CI level exceeds 0% or if CI level is 100%. Furthermore, the infection frequency can only increase toward this equilibrium value if it first reaches a threshold frequency, the level of which also depends upon these three parameters.

View larger version (32K): In this window In a new window Download PPT slide   Figure 1. Cytoplasmic incompatibility. Infection statuses of parents and offspring are indicated in circles. Crosses () symbolize embryonic mortality. (A) Unidirectional incompatibility. Infected females are fully fertile when mating with infected (w) as well as uninfected () males, while embryonic mortality occurs when uninfected females mate with infected males. (B) Bidirectional incompatibility. When males and females are infected, crosses are compatible only if the two partners bear the same Wolbachia variant.

The mechanism of CI induction is currently unknown. However, the mod resc model allows interpretation of the various patterns observed so far (WERREN 1997 ). It postulates the existence of two bacterial functions: mod (for modification) and resc (for rescue). The mod function acts on the nucleus in the males' germline, before Wolbachia are shed from maturing sperm (PRESGRAVES 2000 ). If sperm is affected by mod, zygote development will fail unless resc is expressed in the egg.

Interestingly, CI is also observed in crosses between infected males and infected females, when the two partners bear different Wolbachia strains (O'NEILL and KARR 1990 ). In such cases, CI occurs in both directions of cross and is thus termed bidirectional (Fig 1B). Bidirectional CI demonstrates that mod and resc interact in a specific manner. Two Wolbachia strains are compatible with each other only if they harbor the same compatibility type, defined by a given mod resc pair. Two hypotheses can be proposed to account for the existence of different compatibility types. First, CI might have emerged many times independently, giving rise to different independent mod resc pairs. Alternatively, the different CI systems existing today might derive from one or a few ancestral ones, in which case bidirectionally incompatible strains must have evolved from compatible ancestors. This second hypothesis should be preferred, because it is far more parsimonious. This leaves a problem to solve: How can new compatibility types evolve? This article provides insights into this question.

	ARE MOD AND RESC CONTROLLED BY THE SAME GENES?

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

A biochemical model has been proposed, according to which mod and resc are controlled by the same genetic determinant(s) (CALLAINI et al. 1997 ). It is out of the scope of this article to discuss in depth the validity of this model, but let us consider its theoretical consequences on the evolution of compatibility types. If mod and resc are controlled by the same determinant(s), no asymmetrical changes can occur between the two functions. As a consequence, any mod resc mutant is necessarily self-compatible and bidirectionally incompatible with the original strain (fully or only partially). Previous models on the dynamics of bidirectionally incompatible strains showed that a variant cannot invade when rare (ROUSSET et al. 1991 ; FRANK 1998 ). Thus, if mod and resc are controlled by the same determinant(s), new compatibility types cannot invade, unless selection is counteracted by stochastic events. One might suggest that the spread of such mod resc mutants is facilitated if the mutant clones are at the same time advantaged in terms of transmission efficiency and/or fitness effects to the host (a similar, but not strictly identical, proposition is given in TURELLI 1994 ). However, there is no a priori reason to think that mutations affecting compatibility types should also affect transmission efficiency and/or fitness effects.

Actually, some empirical evidence suggests that different genes control the mod and resc functions. Indeed, some Wolbachia strains that are unable to induce CI but are capable of rescuing it were discovered (BOURTZIS et al. 1998 ; MERCOT and POINSOT 1998 ; POINSOT and MERCOT 1999 ). This finding strongly suggests that mod and resc are genetically separate: if not different genes, at least different gene domains. WERREN 1998 , discussing the process involved in the evolution of compatibility types, assumed that asymmetrical changes could occur between mod and resc. Thus, although not explicitly stated, mod and resc are considered as genetically separate. Werren argued that the emergence of a new compatibility type can occur through an intermediate stage, involving a mutant able to rescue its own CI as well as the one induced by the resident bacterium. If mod and resc are considered independently, two mutations are necessary for such a bacterium to emerge: (i) one change in the mod function (making the original strain unable to rescue the CI induced by the mutant bacterium) and (ii) one change in the resc function, allowing the mutant bacterium to rescue both its own CI and the original strain's one. Such double mutations are highly unlikely. As a consequence, Werren's explanation (in its present form and following our interpretation) is not fully satisfactory. We describe below a process that allows the emergence of new compatibility types under a wider range of conditions, which is based on the hypothesis that mod and resc are genetically separate.

	NOTATIONS AND ASSUMPTIONS

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For the purpose of this article, Wolbachia strains are defined by four parameters: mod compatibility (mod_C), mod intensity (mod_I), resc compatibility (resc_C), and resc intensity (resc_I). mod_C and resc_C are qualitative traits that define the compatibility type. mod_I is a quantitative trait referring to the frequency of embryo death in incompatible crosses. mod_I can vary from 0 (CI level = 0%) to 1 (CI level = 100%). Finally, resc_I is a quantitative trait referring to the frequency of rescued embryos when the compatibility between mod_C and resc_C is complete. resc_I can vary from 0 (nonfunctional resc) to 1 (fully functional resc).

To illustrate our notation, let us describe the following strain, referred to as "strain 0" (S0) in the sections below. Its properties are noted as follows: For M_A,yR_A,z, M refers to mod; the two subscripts refer to mod_C (capital letter) and mod_I (small letter), respectively. R refers to resc; the two subscripts give resc_C (capital letter) and resc_I (small letter), respectively. A given mod_C is compatible with a given resc_C if M and R bear the same capital subscript (i.e., M_A,y is compatible with R_A,x, R_A,y, or R_A,z). Thus, in subscripts, capital letters refer to qualitative traits (A or B in the sections below, with M_A M_B and R_A R_B), and small letters refer to quantitative traits (x or y or z in the sections below, with 0 M_x < M_y < M_z 1 and 0 R_x < R_y < R_z 1).

We analyze the emergence of new compatibility types under the following list of assumptions:

1. Any mutation affecting mod_C or resc_C renders these two totally incompatible (no partial compatibility).

2. As previously mentioned, we suppose that mod (i.e., mod_C + mod_I) is independent from resc (i.e., resc_C + resc_I). Furthermore,

3. mod_I is independent from mod_C, as well as resc_I from resc_C.

4. Mutations affecting mod and resc do not interfere with the efficiency of maternal transmission or the effect of Wolbachia on host fitness (although maternal transmission might not be perfect and Wolbachia might have an effect on host fitness).

5. Recombination between Wolbachia strains cannot occur.

6. A given individual host is homogeneous with regard to Wolbachia infections (when a mutation gives rise to a new clone, its host is infected by this clone only). Finally,

7. host populations are considered as panmictic,

8. with unbiased sex ratio, and

9. nonoverlapping generations.

The results discussed below are qualitatively robust to relaxing assumptions 1 and 3 (data not shown).

	EVOLUTIONARY FORCES ACTING ON MOD AND RESC VARIATIONS

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Variations of resc:
Let us first discuss the probable fate of variations affecting the resc function. Consider a host population (population 1, harboring a unique Wolbachia strain S0, M_A,yR_A,z; Fig 2) and a strain S1 (M_A,yR_A,y, with R_y < R_z) arising by a mutation affecting the resc_I function of an S0 bacterium (Fig 2). S1 is selected against since females bearing S1 suffer a fertility deficit when mating with males infected by S0 or S1. Similarly, a strain S2 (M_A,yR_B,z, with R_B R_A) arising by a mutation affecting the resc_C function of an S0 bacterium (Fig 2), would be eliminated. Indeed, as illustrated in Fig 3A, females bearing S2 are not fully fertile when mating with males infected by S0 or S2. Thus, the efficiency of resc is expected to be optimized: Any reduction of resc_I or change in resc_C is limited by selection. As a consequence, the evolution of new compatibility types cannot start from changes in the resc function. As discussed below, variations affecting the mod function are far less constrained.

View larger version (32K): In this window In a new window Download PPT slide   Figure 2. Identity of the different Wolbachia variants and mutational relationships between them. New mutations are underlined.

View larger version (27K): In this window In a new window Download PPT slide   Figure 3. Patterns of compatibility when rescC or modC are affected. Infection statuses of parents and offspring are indicated in circles. Crosses () symbolize embryonic mortality. For simplicity, modI and rescI are not shown. (A) Patterns of compatibility between S0 and S2. Note that females bearing S2 suffer a fertility deficit when mating with both types of males, so that S2 is counterselected. (B) Patterns of compatibility between S0 and S4. Note that females bearing S0 and females bearing S4 show the same compatibility patterns, so that S0 and S4 have the same fitness.

Variations of mod:
Consider population 1 (infected by S0, M_A,yR_A,z) and a strain S3 (M_A,xR_A,z, with M_x < M_y), arising by a mutation affecting the mod_I function of an S0 bacterium (Fig 2). In crosses involving infected males and uninfected females, S3 will induce a lower CI than S0. As a consequence, the overall infection frequency will decrease (indeed, if maternal transmission is not perfect, the infection frequency at equilibrium depends on CI level). However, given that resc is not affected, females bearing S3 and females bearing S0 are equally compatible with all types of males. Thus, S3 and S0 have the same fitness: As previously stated (PROUT 1994 ; TURELLI 1994 ), variations of mod_I are neutral. Note, however, that such a conclusion has to be tempered if the host population is structured. Indeed, in structured populations, high CI levels are selected for through a kin selection process (FRANK 1997 ).

What about variations affecting mod_C? Consider population 1 (infected by S0, M_A,yR_A,z) and a strain S4 (M_B,yR_A,z, with M_B M_A) arising by a mutation affecting the mod_C function of an S0 bacterium (Fig 2). As illustrated in Fig 3B, fertility is reduced in crosses between males bearing S4 and females bearing S4 or S0. However, given that the resc function did not change, females bearing S4 or S0 are equally compatible with all types of males. Thus, S4 and S0 have the same fitness: Variations of mod_C are neutral. This provides conditions for the emergence of new compatibility types, which we now consider.

	THE EMERGENCE OF NEW COMPATIBILITY TYPES

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Consider a host population (population 2, harboring S0, M_A,yR_A,z, and S4, M_B,yR_A,z). The relative proportion of these two bacterial variants changes through genetic drift only. Consider a strain S5 (M_B,yR_B,z, with R_B R_A), self-compatible, arising by a mutation affecting the resc_C function of an S4 bacterium (Fig 2). As illustrated in Fig 4A, S5 is counterselected if the frequency of M_A variants exceeds that of M_B variants, that is, if f(S0) > f(S4 + S5). In contrast, S5 will invade the population deterministically if f(M_B) > f(M_A) (Fig 4B). Simply speaking, the bacteria selected for are the ones bearing the resc_C function compatible with the most frequent mod_C function. Thus, provided that drift resulted in f(S4) exceeding f(S0), any S5 strain will deterministically invade the population, leading to a shift of compatibility type from M_AR_A to M_BR_B. Let us emphasize that this process does not imply several simultaneous mutational events. Note also that, at any time, natural populations are likely to be polymorphic with regard to mod_C, given that variations are neutral. If several mod_C functions coexist, a new compatibility type will invade as soon as the appropriate mutation affecting resc_C occurs in a Wolbachia bearing the most frequent mod_C function.

View larger version (16K): In this window In a new window Download PPT slide   Figure 4. Fate of S5 when occurring in population 2 (harboring S0 and S4). These numerical examples were obtained with the following conditions: My = 1; Rz = 1; overall infection frequency = 1; perfect maternal transmission; no cost to the host. Algebraic details are in Appendix A. (A) Initial situation 1: f(MB) < f(MA). f(S0) = 0.6; f(S4) = 0.3; f(S5) = 0.1. (B) Initial situation 2: f(MB) > f(MA). f(S0) = 0.3; f(S4) = 0.6; f(S5) = 0.1.

Invasion by a new compatibility type may be facilitated by mutations affecting mod_I. Indeed, consider population 2 (harboring S0, M_A,yR_A,z, and S4, M_B,yR_A,z) and a strain S6 (M_B,zR_A,z, with M_z > M_y) arising by a mutation affecting the mod_I function of an S4 bacterium (Fig 2). In such a population (population 3, harboring S0, S4, and S6), the relative proportion of the three variants changes through genetic drift only. If S5 (M_B,yR_B,z) occurs in population 3, it may invade the population even if f(M_B) < f(M_A), as illustrated in Fig 5A. The bigger the difference between M_z and M_y, the lower the frequency of M_B that must be reached for S5 to invade deterministically.

View larger version (19K): In this window In a new window Download PPT slide   Figure 5. Consequences of mod intensity variations on the emergence of new compatibility types. These numerical examples were obtained with the following conditions: My = 0.6; Mz = 1; Rz = 1; overall infection frequency = 1; perfect maternal transmission; no cost to the host. Algebraic details are in Appendix B. (A) S2 occurs in a population harboring S0, S4, and S6. In the initial situation, f(S0) = 0.59; f(S4) = 0.05; f(S6) = 0.31; f(S5) = 0.05. Note that S5 invades although f(MB) < f(MA), because Mz > My. (B) S7 occurs in a population harboring S0, S4, and S6. In the initial situation, f(S0) = 0.59; f(S4) = 0.05; f(S6) = 0.31; f(S7) = 0.05. Note that S7 invades although f(MB) < f(MA), because Mz > My.

Interestingly, the process involved in the shift to a new compatibility type might also lead to an overall increase of CI levels. Indeed, consider population 3, harboring S0 (M_A,yR_A,z), S4 (M_B,yR_A,z), and S6 (M_B,zR_A,z). Consider now that instead of S5 (M_B,yR_B,z, bearing M_y), a strain S7 (M_B,zR_B,z, with R_B R_A) arises by a mutation affecting the resc_C function of an S6 bacterium (Fig 2). This strain invades population 3 in the same general conditions as S5, as described in the above paragraph, although more rapidly (Fig 5B). However, in the present case, the CI level is finally higher than in the previous situation, given that M_z > M_y. Thus, the process involved in the evolution of compatibility types might not simply be facilitated by mutations increasing mod_I; it might also induce by itself an increase of CI level. Higher transmission efficiency or lower cost to the host might also favor the spread of new compatibility types. However, these two parameters are expected to be optimized by selection in natural populations (TURELLI 1994 ) so that mutants with increased transmission efficiency or decreased cost to the host are less likely to appear than mutants with increased CI level.

	EVOLUTION AND STABLE MAINTENANCE OF SUICIDAL WOLBACHIA

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Consider population 2 (harboring S0, M_A,yR_A,z, and S4, M_B,yR_A,z) and the strain S2 (M_A,yR_B,z, with R_B R_A) arising by a mutation affecting the resc_C function of an S0 bacterium (Fig 2). Remember that S2, when arising in population 1 (infected by S0 only) is selected against. Different outcomes may occur in population 2. As illustrated in Fig 6A, S2 is lost if f(M_A) > f(M_B), that is, if f(S0 + S2) > f(S4). In contrast, the S2 frequency will increase if f(M_A) < f(M_B). Indeed, if f(M_A) < f(M_B), S2 bears the resc_C function compatible with the most frequent mod_C. As f(S2) increases, f(M_B) decreases and f(M_A) increases, until f(M_A) = f(M_B); that is, f(S0 + S2) = f(S4), which is a stable equilibrium (Fig 6B). The population (population 4, harboring S0, S2, and S4) thus presents a stable polymorphism of Wolbachia strains: one self-compatible strain (S0) and two "suicidal" strains (S4 and S2), unable to rescue their own CI phenotype, but able to rescue the one induced by another strain. This polymorphism is stable in that any deviations of frequencies are limited by selection. However, note that the equilibrium might be broken if an S5 (M_B,yR_B,z) strain occurs in population 4, as S5 could invade the population.

View larger version (15K): In this window In a new window Download PPT slide   Figure 6. Fate of S2 when occurring in population 2 (harboring S0 and S4). These numerical examples were obtained with the following conditions: My = 1; Rz = 1; overall infection frequency = 1; perfect maternal transmission; no cost to the host. Algebraic details are in Appendix C. (A) Initial situation 1: f(MB) < f(MA). f(S0) = 0.6; f(S4) = 0.3; f(S2) = 0.1. (B) Initial situation 2: f(MB) > f(MA). f(S0) = 0.2; f(S4) = 0.7; f(S2) = 0.1.

	GENERALIZATION OF THE RESC FUNCTION

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Consider population 2 (harboring S0, M_A,yR_A,z, and S4, M_B,yR_A,z) and a strain S8 (M_A,yR_AB,z, with R_AB R_A) arising by a mutation affecting the resc_C function of an S0 bacterium, or a strain S9 (M_B,yR_AB,z, with R_AB R_A) arising by a mutation affecting the resc_C function of an S4 bacterium (Fig 2). S8, as well as S9, bears a resc_C function compatible both with M_A and M_B. Such strains are selected for, regardless of f(M_A) and f(M_B). In other words, generalization of resc_C is always selected for. Interestingly, POINSOT et al. 1998 reported the case of a Wolbachia strain able to rescue two different mod functions, suggesting the existence of such super-resc functions. If S8 gets fixed, selection on R_B is relaxed, which might eventually lead to its loss. Similarly, if S9 gets fixed, R_A might eventually be lost, leading to a shift of compatibility type from M_AR_A to M_BR_B.

This process, involving an intermediate Wolbachia strain harboring a specific mod_C and a "double" resc_C, can be compared to WERREN's (1998) hypothesis. It is important to stress two original facets of the present proposition, making it more satisfactory: (i) The two mutational events do not have to be simultaneous, since variations of mod_C are neutral, and (ii) for a shift of compatibility type to occur, there is no need that two mutations leading to a double resc_C function occur in a different manner and in different populations (isolated in space or in time).

	CONSEQUENCES ON HOST MEAN FITNESS

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CI can affect host population mean fitness in various ways. First, CI-inducing Wolbachia may be costly to their host (negative effect on host fitness) and yet be maintained at high frequencies through the effect of CI (CASPARI and WATSON 1959 ). Second, when infection is not fixed, a proportion of crosses within the population are incompatible. Finally, the occurrence of suicidal Wolbachia strains can greatly affect population mean fitness. Any population harboring non-self-compatible strains suffers a mean fitness reduction because of these latter. As an example, population 2, harboring S0 (M_A,yR_A,z) and S4 (M_B,yR_A,z) suffers a mean fitness reduction owing to the presence of the S4 strain (see also Fig 4A and Fig 6A, where the mean fitness is much lower than 1, because of S4). In population 2, S4 frequency varying under drift, the population can eventually go extinct if S4 gets fixed (at fixation, mean fitness = 0 if M_y = 1). The stable equilibrium described above (population 4, harboring S0, S2, and S4) is also interesting in this respect. Indeed, as illustrated in Fig 6B, population mean fitness is fixed to 0.5 at equilibrium: On average, half of the eggs do not hatch because of CI (if M_y = 1).

Mean fitness reductions of this magnitude are very likely to affect population demography and might render suicidal strains rare, through the extinction of populations bearing them. If suicidal mutants occur frequently, Wolbachia-infected populations might indeed go extinct frequently because of reduced mean fitness. The actual consequences of embryonic mortality caused by CI on population viability will depend on the type of ecological factors limiting population size. If population size is limited mainly by density-dependent factors, such as competition for food, the population demography is likely to be less affected than if population size is limited mainly by non-density-dependent factors. Consequently, Wolbachia infections might be rarer in species where population size is limited mainly by non-density-dependent factors.

	CONCLUSION AND PROSPECTS

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Our analysis suggests that if mod and resc are genetically separate, new compatibility types are likely to evolve under a wide range of conditions, through a process involving drift and selection. This being so, compatibility types cannot be considered as evolutionarily stable in finite populations. Generalization of the resc function might represent an intermediate stage in the evolution of new compatibility types, although it is not an indispensable step. Finally, we have shown that stable polymorphism can be maintained, allowing the long-term existence of suicidal Wolbachia strains, with heavy consequences on population mean fitness.

For this analysis, we assumed that when a mutation occurs, the individual host is infected by the mutant clone only [assumption (6)]. The underlying hypothesis is that the effective bacterial population size is very small within an individual host. This assumption might be justified if Wolbachia clones get through tight bottlenecks at every generation, during the germ cells' colonization within the developing embryo. Yet, multiple infections are stably maintained in natural populations (MERCOT et al. 1995 ; ROUSSET and SOLIGNAC 1995 ; WERREN et al. 1995 ), suggesting that population size is not that small. Double infections can even be maintained for many generations in experiments where selection for the presence of both strains is relaxed (POINSOT et al. 2000 ). Taking this fact into consideration might reveal interesting features with regard to the evolution of compatibility types.

Future models concerned with the evolution of mod and resc will undoubtedly have to include nondeterministic processes, as these seem to play a fundamental role. Simulation programs, combining the effects of mutation, selection, and drift, should tell us how plausible are the different outcomes described here. Empirical tests are also required. In particular, the rate at which bidirectional incompatibility evolves must be estimated. For now, complete bidirectional incompatibility has been reported only from evolutionarily distant strains. It should not be hastily concluded from this (lack of) observation that the evolution of compatibility types is a slow process, given that only very few closely related strains have been confronted. This issue could be more deeply investigated through artificial injections of several Wolbachia strains, more or less closely related, within a single host. Finally, if the suicidal Wolbachia is to be found, it will come from the field.

	ACKNOWLEDGMENTS

We thank Greg Hurst, Denis Poinsot, Hadi Quesneville, Fabrice Vavre, and the two anonymous reviewers for helpful comments on previous versions of this article.

Manuscript received April 10, 2001; Accepted for publication September 17, 2001.

	APPENDIX A

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

ALGEBRAIC DETAILS FOR Fig 4
The fitness of the different variants is the probability that females bearing them mate with compatible males. If f(S0) = P, f(S4) = Q, and f(S5) = R, then

and the population mean fitness is

The frequencies of the different variants at generation N + 1 are functions of the frequencies at generation N:

	APPENDIX B

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

ALGEBRAIC DETAILS FOR Fig 5
Fig 5A:
If f(S0) = P, f(S4) = Q, f(S6) = T, and f(S5) = R, then

and the population mean fitness is

The frequencies of the different variants at generation N + 1 are functions of the frequencies at generation N:

Fig 5B:
If f(S0) = P, f(S4) = Q, f(S6) = T, and f(S7) = R, then

and the population mean fitness is

The frequencies of the different variants at generation N + 1 are functions of the frequencies at generation N:

	APPENDIX C

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

ALGEBRAIC DETAILS FOR Fig 6
If f(S0) = P, f(S4) = Q, and f(S2) = R, then

and the population mean fitness is

The frequencies of the different variants at generation N + 1 are functions of the frequencies at generation N:

	LITERATURE CITED

TOP ABSTRACT ARE MOD AND RESC... NOTATIONS AND ASSUMPTIONS EVOLUTIONARY FORCES ACTING ON... THE EMERGENCE OF NEW... EVOLUTION AND STABLE MAINTENANCE... GENERALIZATION OF THE RESC... CONSEQUENCES ON HOST MEAN... CONCLUSION AND PROSPECTS APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

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