Nicky Wybouw

Emma Van Reempts

Jens Zarka

Flore Zélé

Dries Bonte

Emma Van Reempts

Egg provisioning explains the penetrance of symbiont-mediated sex allocation distortion in haplodiploids

Egg provisioning explains the penetrance of symbiont-mediated sex allocation distortion in haplodiploids

Introduction

Maternally inherited symbionts can spread rapidly within arthropod populations by inducing a variety of reproductive phenotypes in their host [START_REF] Engelstädter | The Ecology and Evolution of Microbes that Manipulate Host Reproduction[END_REF]. These reproductive phenotypes increase the proportion of infected females that transmit the symbiont in host populations. Cytoplasmic incompatibility (CI) is currently recognized as the most common reproductive manipulation. CI interferes with the development of uninfected host embryos that were fertilized by infected males [START_REF] Beckmann | The Toxin-Antidote Model of Cytoplasmic Incompatibility: Genetics and Evolutionary Implications[END_REF][START_REF] Shropshire | Symbiont-mediated cytoplasmic incompatibility: what have we learned in 50 years?[END_REF]. In arthropod hosts with a haplodiploid mode of reproduction (unfertilized eggs develop into haploid males and fertilized eggs into diploid females), fertilized eggs that suffer from symbiontinduced CI have two developmental outcomes. Embryos either die before reaching adulthood (Female Mortality CI (FM-CI)) or develop into viable males (Male Development CI (MD-CI)) [START_REF] Vavre | Within-species diversity of Wolbachia-induced cytoplasmic incompatibility in haplodiploid insects[END_REF][START_REF] Perrot-Minnot | Contrasting effects of Wolbachia on cytoplasmic incompatibility and fecundity in the haplodiploid mite Tetranychus urticae: Contrasting effects of Wolbachia in a haplodiploid mite[END_REF] (Figure 1). Previous work showed that a single CI cross in haplodiploids can simultaneously result in FM-CI and MD-CI [START_REF] Vavre | Within-species diversity of Wolbachia-induced cytoplasmic incompatibility in haplodiploid insects[END_REF]Wybouw et al., 2022) (Figure 1). As CI is rescued when males and females have an identical infection state, the reproductive phenotype provides a selective advantage to transmitting females. In haplodiploid arthropods, parthenogenesis induction is another classic and wellstudied invasion strategy of symbionts [START_REF] Ma | Patterns and mechanisms in instances of endosymbiont-induced parthenogenesis[END_REF]. In contrast to CI, symbiontinduced parthenogenesis does not rely on egg fertilization and causes a diploidization of unfertilized eggs, generating parthenogenetic females instead of males that do not transmit the symbiont [START_REF] Weeks | Wolbachia -induced parthenogenesis in a genus of phytophagous mites[END_REF][START_REF] Ma | Patterns and mechanisms in instances of endosymbiont-induced parthenogenesis[END_REF]. Another, much less understood, strategy of maternally transmitted symbionts to increase the proportion of infected females in haplodiploid host populations consists of directly altering sex allocation (i.e. the parental investment of resources to female vs. male offspring).

As for CI, egg fertilization by males is essential for sex allocation distortion. However, in contrast to CI, sex allocation distortion occurs in crosses with infected females and is not contingent on the infection state of males (Figure 1). These changes in sex allocation result in female-biased broods of infected females compared to those of uninfected females. To our knowledge, this reproductive phenotype was first described in a strain of the haplodiploid spider mite Tetranychus urticae that was infected with Wolbachia [START_REF] Vala | Wolbachia -induced 'hybrid breakdown' in the twospotted spider mite Tetranychus urticae Koch[END_REF][START_REF] Vala | Genetic Conflicts over Sex Ratio: Mite-Endosymbiont Interactions[END_REF]. In addition to Wolbachia, sex allocation distortion is now associated with other bacterial symbionts. Rickettsia, Hamiltonella, and Arsenophonus influence offspring sex ratio of infected females in haplodiploid whitefly species [START_REF] Himler | Rapid Spread of a Bacterial Symbiont in an Invasive Whitefly Is Driven by Fitness Benefits and Female Bias[END_REF][START_REF] Shan | The inherited bacterial symbiont Hamiltonella influences the sex ratio of an insect host[END_REF][START_REF] Wang | Intracellular symbionts drive sex ratio in the whitefly by facilitating fertilization and provisioning of B vitamins[END_REF]. Similarly, Cardinium also influence sex allocation in infected Kelly's citrus thrips Pezothrips kellyanus in favour of female offspring [START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. Haplodiploid arthropods, including Tetranychus mites and Pezothrips insects, allocate egg fertilization and female development to larger eggs [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF]Katlav, Cook, et al., 2021). [START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF] show that Cardinium affect sex allocation in P. kellyanus by increasing egg size investment, hereby facilitating egg fertilization (and thus female development). Cardiniummediated increase in egg size also led to progeny that were characterized by a higher survival rate and larger adult body size. In contrast, we do not understand how Wolbachia induce sex allocation distortion. The host genotype modulates the expression of many Wolbachia-induced reproductive phenotypes [START_REF] Fujii | Transfection of Wolbachia in Lepidoptera: the feminizer of the adzuki bean borer Ostrinia scapulalis causes male killing in the Mediterranean flour moth Ephestia kuehniella[END_REF][START_REF] Kaur | Living in the endosymbiotic world of Wolbachia: A centennial review[END_REF]Wybouw et al., 2022), raising the question whether sex allocation distortion is also contingent on host genetics.

Unravelling the mechanistic underpinnings of Wolbachia-induced sex allocation distortion is not only crucial for our fundamental understanding of host-Wolbachia interactions but also for the development of integrated pest management that relies on Wolbachia. A number of pest management programs take advantage of the CI drive system of Wolbachia to replace native pest populations with Wolbachia-infected populations that carry favourable traits, such as reduced virus transmission [START_REF] Ross | Evolutionary Ecology of Wolbachia Releases for Disease Control[END_REF]. However, CI strength can be weakened by varying population and environmental effects [START_REF] Ross | Wolbachia Infections in Aedes aegypti Differ Markedly in Their Response to Cyclical Heat Stress[END_REF]Wybouw et al., 2022), threatening a collapse of the pest control technique. Previous theoretical work predicts that sex allocation distortion could reinforce the effect of CI and promote the invasion of Wolbachia from low initial infection frequencies [START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF]. Yet, the actual contribution of sex allocation distortion to the spread of Wolbachia in natural pest populations remains unclear. Indeed, this previous model was not parametrized using empirical data of Wolbachia variants that cause both CI and sex allocation distortion. Moreover, in conflict with recent findings (Wybouw et al., 2022), the previous model did not allow for a simultaneous expression of FM-CI and MD-CI in single CI crosses. Understanding whether and how sex allocation of haplodiploid pests can be altered during population replacement events is critical to predict the outcome of Wolbachia-based pest control.

Here, we use the haplodiploid Tetranychus system to study a Wolbachia variant that simultaneously induces CI and sex allocation distortion. We show that sex allocation distortion is determined by the host background using a nuclear genotype reference panel of T. urticae. We further examined whether this reproductive phenotype is associated with changes in oviposition and immature survival rate. We tested whether, in analogy with Cardinium, Wolbachia also control sex allocation of haplodiploid hosts via egg size provisioning, and whether sex allocation distortion is associated with changes in adult size. Our results reveal signatures of convergent mechanisms of sex allocation distortion across two divergent host-This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 symbiont systems. Finally, we parametrized a mathematical model to determine whether our observed levels of sex allocation distortion would contribute to the invasion of Wolbachia in natural haplodiploid populations.

Materials and methods

Mite lines and maintenance

We selected six T. urticae near-isogenic lines from the nuclear genotype reference panel described in Wybouw et al., 2022;Beis-w, Beis-c, LonX-w, LonX-c, Stt-w, and Stt-c. These lines are composed of three different near-isogenic nuclear genotypes (Beis, LonX, and Stt) that were transferred by seven rounds of iterative paternal introgression into a single cytoplasm that was originally recovered from the Wolbachia-infected Scp-w line. All lines share a single mitochondrion and were either infected with a single CI-inducing Wolbachia variant (indicated by "-w") or were cured of the infection by antibiotic treatment (indicated by "-c") (Wybouw et al., 2022). After antibiotic curing, the cured lines were propagated for ~18 generations before the onset of the experiments. Selection of the three nuclear genotypes from the reference panel was based on CI strength and phenotype. In addition, we also used the original Stt nearisogenic line that was naturally uninfected and did not undergo any antibiotic treatment. Infection states of Wolbachia and other maternally transmitted symbionts were confirmed before the start of the experiments by diagnostic PCR assays, as previously described (Wybouw et al., 2022). Mite lines were propagated by serial passage on detached bean leaves (Phaseolus vulgaris L. cv 'Prelude') at 24°C, 60% RH, and a 16:8 light:dark photoperiod.

Effects of Wolbachia on sex allocation

To obtain estimates of sex ratio and immature stage survival (ISS), we generated age cohorts of the mite lines by allowing 75 randomly collected adult females to oviposit for 24 hours on a detached bean leaf. At the end of offspring development (between nine and eleven days postoviposition), ~50 females were isolated on a single day as teleiochrysalids (the last immobile molting stage before reaching adulthood). These females were paired with ~30 one-to threeday old adult males of their own line on a detached bean leaf, allowing for compatible withinline matings. Three days were given to the females to emerge as virgin adults, and for matings to occur (we confirmed that the near-isogenic lines exhibit high copulation propensity under laboratory conditions; Supplementary information Box S1). Replicates were established by pooling five females (males were discarded) and allowing oviposition on 16 cm 2 leaf discs for three days. Females were removed from the leaf discs and eggs were counted. Male and female offspring were collected and counted upon reaching adulthood. For Beis-w, Beis-c, LonX-w, LonX-c, Stt-w, and Stt-c, a total of 14 replicates were performed per line across three different batches (experimental blocks, each corresponding to a different age cohort). To fully ensure that the observed decrease in sex ratio in Stt-c vs. Stt-w was not associated with longterm deleterious effects of the antibiotic treatment, we expanded the experimental set-up for this host genotype by adding the original uninfected near-isogenic Stt line. Here, following the same rearing and mating protocol as described above, five replicates for Stt (uninfected), Sttw (infected), and Stt-c (cured) were performed in a single batch.

To verify that the effect of Wolbachia on the sex ratio of Stt-w is not caused by Wolbachia-mediated parthenogenesis, feminization, or male killing, we isolated ~50 teleiochrysalid females from age cohorts of Stt-c and Stt-w as described above. We allowed the teleiochrysalids to hatch into virgin adult females and feed on a detached bean leaf for three days. As above, replicates were established by pooling five females on 16 cm 2 leaf discs and allowing oviposition for three days. Females were removed from the leaf discs and eggs This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 were counted. Male and female offspring were collected and counted upon reaching adulthood. For both Stt-c and Stt-w, a total of nine replicates were performed across two different batches.

Effects of Wolbachia on offspring number and size

To obtain accurate estimates of oviposition rates, we isolated ~50 teleiochrysalid females from age cohorts of Beis-w, Beis-c, LonX-w, LonX-c, Stt-w, and Stt-c, created as described above. These females were paired with ~30 one-to three-day old adult males on a detached bean leaf and were allowed to mate for three days. Replicates were established by pooling two females and allowing oviposition on 16 cm 2 leaf discs for 24 hours. Two hours after creating the replicates, all experimental females were examined to ensure they survived the transfer. Replicates with one or two dead females were removed from further analyses. After the oviposition interval, females were removed from the leaf discs and eggs were counted. The number of replicates per mite line ranged between 25 and 43 and were performed across four different batches.

To determine the effect of Wolbachia infection on Tetranychus egg size, eggs were randomly sampled from the leaf discs that were used to estimate oviposition rates for Beis-w, Beis-c, LonX-w, LonX-c, Stt-w, and Stt-c. Here, we focused on eggs that were oviposited across three batches of the oviposition tests. Eggs were photographed under a binocular microscope (6x 10x) using a Leica M50 camera (5 MP HD Microscope Camera Leica MC170 HD). Between 125 and 197 eggs were analyzed for each mite line (with a total of 971 eggs). To confirm that sex allocation is mediated by egg size in T. urticae [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF], between 21 and 37 photographed eggs from Beis-c, Beis-w, LonX-c, and LonX-w were isolated and transferred to separate 4 cm 2 leaf discs (with a total of 124 eggs). Sex was determined at the sexually dimorphic teleiochrysalid stage. Photos were analyzed using Natsumushi image measuring software [START_REF] Tanahashi | Natsumushi : Image measuring software for entomological studies: Image measuring software for entomology[END_REF]. For each egg, Natsumushi generated a smoothed outline based on eight manually specified demarcating points and automatically extracted the projected egg area. The total egg surface was calculated by assuming a spherical shape, following previous studies [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF][START_REF] Macke | Mating Modifies Female Life History in a Haplodiploid Spider Mite[END_REF].

To determine the effect of Wolbachia infection on Tetranychus adult size, ~75 adult females were isolated from age cohorts of LonX-w, LonX-c, Stt-w, and Stt-c, and allowed to oviposit on a detached bean leaf for 24 hours. Larval offspring were collected upon hatching and 12 larvae were pooled on 16 cm 2 leaf discs to ensure equal population densities. As adult mites are highly mobile and are difficult to position in equal planes, we used the projected body surface of teleiochrysalids as a proxy for adult size. Both male and female teleiochrysalids were isolated, placed on a flat perspex slide, and photographed as described above. Using Natsumushi image measuring software [START_REF] Tanahashi | Natsumushi : Image measuring software for entomological studies: Image measuring software for entomology[END_REF], the projected teleiochrysalid surface was measured by manually specifying ten demarcating points. Between 55 and 66 teleiochrysalids were analyzed for each mite line. In contrast to egg size, adult size was not expressed as total surface, but as the projected surface area.

Statistical analyses

All analyses were carried out using R (version 4.1.3) (R Core Team, 2021). Raw data and the R script are publicly accessible as Supplementary Files and the statistical models are described in Table S1. Sex ratio was calculated as the proportion of females among the progeny that successfully reached adulthood. Immature stage survival (ISS) included embryonic and juvenile survival and was defined as the proportion of adult offspring over the total number of eggs. Daily oviposition was computed as the number of eggs per alive female This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 (oviposition was allowed for 24 hours only). All data, except copulation latency (Supplementary information Box S1), were analyzed using generalized linear mixed models (glmmTMB function of the "glmmTMB" package) [START_REF] Brooks | glmmTMB balances speed and flexibility among packages for zero-inflated generalized linear mixed modeling[END_REF]. Proportional data, sex ratio and ISS, were examined with a Binomial (or Betabinomial to account for overdispersion) error distribution, with a logit link. Continuous data, oviposition, egg surface, male and female adult size were fitted with a Gaussian error distribution (Table S1). In the case of non-normal error distribution, the response variables were linearized using a Box-cox transformation prior to analysis (boxcox function of the "MASS" package was used to find the appropriate value of lambda) [START_REF] Crawley | The R book[END_REF]. For most of the statistical models, genotype, infection state and their interaction were fit as fixed explanatory variables, whereas batch was considered as a random explanatory variable. To analyze sex ratio within the Stt background (Stt, Stt-c, and Stt-w), only the infection state was used as a fixed explanatory variable. For the analysis of female and male egg surface, offspring sex along with all two-and three-way interactions with the genotype and/or infection state were added as fixed explanatory variables. Maximal models, including all higher order interactions, were simplified by sequentially eliminating nonsignificant terms and interactions to establish minimal models. The significance of the explanatory variables was established using chi-square tests (Anova function of the "car" package). The reported significant chi-squared values are for the minimal model while the nonsignificant values were obtained before removing the variable from the minimal model [START_REF] Crawley | The R book[END_REF]. When interactions or factors with more than two levels were significant, differences among levels were analyzed using multiple comparisons (pairs function of the "emmeans" package [START_REF] Lenth | emmeans: estimated marginal means, aka least-squares means[END_REF]), with Bonferroni corrections to account for multiple testing.

Modelling Wolbachia invasion in Tetranychus populations

To determine how sex allocation distortion, combined with varying CI phenotype (FM-and/or MD-CI) and strength, may affect the spread of a Wolbachia variant in divergent Tetranychus populations, we extended a mathematical model previously developed by [START_REF] Vavre | Evidence for female mortality in Wolbachia -mediated cytoplasmic incompatibility in haplodiploid insects: Epidemiologic and evolutionary consequences[END_REF][START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF] The proportion of females (𝑓 𝑡 ) and males (𝑚 𝑡 ) that are infected at generation 𝑡, are given by the following equations:

𝑓 𝑡+1 = 𝑓 𝑡 • 𝐹 • 𝐼𝑆𝑆 • 𝑆𝑑 • (1 -µ) 𝑓 𝑡 • 𝐹 • 𝐼𝑆𝑆 • 𝑆𝑑 • (1 -µ • 𝐶𝐼 • 𝑚 𝑡 ) + (1 -𝑓 𝑡 )(1 -𝐶𝐼 • 𝑚 𝑡 ) 𝑚 𝑡+1 = 𝑓 𝑡 • 𝐹 • 𝐼𝑆𝑆 • (1 -µ)(1 -𝑘 • 𝑆𝑑) 𝑓 𝑡 • 𝐹 • 𝐼𝑆𝑆 • (1 -𝑘 • 𝑆𝑑 + µ • 𝑘 • 𝑆𝑑 • 𝑀𝐷 • 𝑚 𝑡 ) + (1 -𝑓 𝑡 )(1 -𝑘 + 𝑘 • 𝑀𝐷 • 𝑚 𝑡 )
where most of the parameters correspond to those in [START_REF] Vavre | Evidence for female mortality in Wolbachia -mediated cytoplasmic incompatibility in haplodiploid insects: Epidemiologic and evolutionary consequences[END_REF]. 𝐹 is the relative fecundity of infected vs. uninfected females, ISS the relative immature stage survival of infected vs. uninfected offspring, and µ the proportion of uninfected eggs produced by infected females (transmission efficiency is thus given by 1-µ). 𝑘 corresponds to the fertilization rate of eggs (i.e., estimated as the female to adult ratio in control crosses), and 𝑆𝑑 to the sex allocation distortion induced by Wolbachia in the brood of infected females (i.e., the ratio of female to adult in the brood of infected females relative to uninfected females in control crosses). Finally, 𝐶𝐼 is the total proportion of eggs affected by CI (i.e., total CI strength), computed as a function of 𝑀𝐷 and 𝐹𝑀, which correspond to the MD-and FM-CI phenotypes, respectively (𝑀𝐷 and 𝐹𝑀 are equal to the corrected indexes 𝑀𝐷 𝑐𝑜𝑟𝑟 and 𝐹𝑀 𝑐𝑜𝑟𝑟 , respectively) [START_REF] Poinsot | Wolbachia Transfer from Drosophila melanogaster into D. simulans: Host Effect and Cytoplasmic Incompatibility Relationships[END_REF][START_REF] Cattel | Back and forth Wolbachia transfers reveal efficient strains to control spotted wing drosophila populations (J Beggs, Ed.)[END_REF]Zélé et al., 2020;[START_REF] Cruz | Wolbachia and host intrinsic reproductive barriers contribute additively to postmating isolation in spider mites[END_REF]Wybouw et al., 2022). Assuming that the effects of both CI phenotypes are independent, total CI should be computed as 𝐶𝐼 = 𝑀𝐷 + 𝐹𝑀 • (1 -𝑀𝐷). In the current study, however, we computed total CI as 𝐶𝐼 = 𝑀𝐷 + 𝐹𝑀, This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 as the implemented 𝐹𝑀 𝑐𝑜𝑟𝑟 indexes already account for a potential decrease in female production due to MD-CI (i.e. these indexes were already corrected using 𝑀𝐷 𝑐𝑜𝑟𝑟 ; Wybouw et al., 2022).

In this model, Wolbachia infection frequency in females is calculated by the same formula as when only MD-or FM-CI is considered [START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF][START_REF] Vavre | Evidence for female mortality in Wolbachia -mediated cytoplasmic incompatibility in haplodiploid insects: Epidemiologic and evolutionary consequences[END_REF] for 𝑆𝑑 = 1). The difference lies in how the 𝐶𝐼 parameter is computed. As only MD-CI directly affects the proportion of infected males, the formula for Wolbachia infection frequency in males is the same as previous models that only consider MD-CI, and when 𝑀𝐷 = 0, it is also the same as those that only consider FM-CI [START_REF] Vavre | Evidence for female mortality in Wolbachia -mediated cytoplasmic incompatibility in haplodiploid insects: Epidemiologic and evolutionary consequences[END_REF] for 𝑆𝑑 = 1). Note, however, that the current formula differs from that of [START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF] for MD-CI, as one category of uninfected males produced by uninfected females, is missing in the denominator of the previous model. The category corresponds to uninfected eggs fertilized by the sperm of infected males that develop as males instead of females due to MD-CI, given by (1 -𝑓 𝑡 )(𝑘 • 𝑀𝐷 • 𝑚 𝑡 ).

As in [START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF]Vavre et al., 2000, when Wolbachia induce CI, the model can yield three different equilibria: (1) an unstable polymorphic equilibrium, which corresponds to 'the threshold infection frequency for Wolbachia invasion' known in diploid species [START_REF] Hoffmann | Factors affecting the distribution of cytoplasmic incompatibility in Drosophila simulans[END_REF][START_REF] Turelli | Evolution of Incompatibility-inducing Microbes and Their Hosts[END_REF]; (2) a stable polymorphic equilibrium, which is reached when the initial infection frequency is above the unstable equilibrium (note that only perfectly transmitted Wolbachia can fully invade a host population; i.e. this equilibrium can equal 1 only when µ = 0), and (3) an equilibrium where Wolbachia infection is purged, which is reached when the initial infection frequency is below the unstable equilibrium. When Wolbachia do not induce CI, only one polymorphic equilibrium can exist that is either stable or unstable when Wolbachia are beneficial or costly for the host, respectively. Analytical solutions and main equilibrial conditions are provided in the Supplementary Box S2.

Results

Sex allocation in Tetranychus relies on an interaction between host and Wolbachia

To study Wolbachia-induced sex allocation distortion in Tetranychus, we selected three nuclear genotypes, Beis, LonX, and Stt that were either infected with a single Wolbachia variant ('-w' suffix) or cured via antibiotic treatment ('-c' suffix) (Wybouw et al., 2022). Selection of these nuclear genotypes was based on varying CI strength and phenotype. Whereas Beis is typified by complete CI, LonX and Stt exhibit intermediate CI strength. The CI phenotype is characterized by a combination of MD-CI and FM-CI in LonX, while only consisting of FM-CI in Beis and Stt (Wybouw et al., 2022). We performed compatible within-line crosses for these Wolbachia-infected and cured sister lines. The interaction between nuclear genotype and the Wolbachia infection state significantly affected the female-biased sex ratios (χ 2 2=14.6286, p=0.0007) (Figure 2). Within-genotype a posteriori comparisons revealed that the average sex ratio increased by ~11% when Stt females were infected with Wolbachia (t-ratio=4.272, p=0.0005), whereas the sex ratios were not significantly different within Beis nor LonX (t-ratio=1.125, p=1.0000 and t-ratio=1.847, p=0.6176, respectively) (Figure 2 and Table S2). In a second experiment, we included the naturally uninfected Stt line to rule out any potential long-term interference of the antibiotic treatment on sex ratio. Infection state determined the female-biased sex ratio (χ 2 2=18.9370, p<0.0001). No significant difference was found between uninfected Stt and cured Stt-c (t-ratio=0.110, p=1.0000), whereas the sex ratio of infected Sttw was higher to both Wolbachia-free lines (vs. Stt-c: t-ratio=3.964, p=0.0056 and vs. Stt: t-ratio=3.864, p=0.0068) (Figure 2). Together with the non-differential sex ratios between the infected and cured sister lines of the Beis and LonX genotype (Figure 2), this is strong evidence that the antibiotic treatment had no long-term effects on observed sex ratios. This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1

Host genotype, infection state, and their interaction did not significantly affect immature stage survival (ISS). ISS is therefore not associated with the change in sex ratio within the Stt background (Figure S1 and Table S3). We also investigated the offspring sex ratio and ISS of virgin Stt-w and Stt-c females to test whether the Wolbachia variant induces parthenogenesis, feminization, or male killing within the Stt background. All offspring were male, ruling out Wolbachia-mediated parthenogenesis and feminization. As no significant difference in ISS was observed between the infected and cured line (χ 2 1=0.0000, p=0.9949), the hypothesis of Wolbachia-induced male killing was also rejected (Figure S1 and Table S3). This further shows that Wolbachia infection does not cause any particular survival costs in the focal nuclear genotypes during offspring development. However, the interaction between host genotype and the Wolbachia infection state significantly determined daily oviposition (χ 2 2=9.0006, p=0.0111) (Figure 3 and Table S4). Comparisons within host nuclear backgrounds uncovered that the daily oviposition rates were only significantly different between the infected and cured line of the LonX genotype (t-ratio=5.704, p<0.0001). For the LonX genotype, Wolbachia infection decreased daily oviposition (Figure 3).

Wolbachia control host sex allocation by regulating egg provisioning

In Tetranychus mites, the probability of egg fertilization (and subsequent female development) positively covaries with egg size [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF]. Here, we confirm that sex allocation is mediated by egg size by tracking the offspring of Beis-w, Beis-c, LonX-w, and LonX-c. Statistical analyses revealed that female egg surface was significantly larger (χ 2 1=28.8573, p<0.0001), and also varied depending on the interaction between nuclear genotype and the infection state (χ 2 3=11.1822, p=0.0008) (Figure S2). In haplodiploid Pezothrips insects, Cardinium manipulate sex allocation by increasing egg size, hereby facilitating egg fertilization [START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. We tested whether Wolbachia achieve higher female-biased sex ratios in Stt mites by inducing similar changes in the host reproductive physiology. We quantified egg size across the infected and cured lines of the three T. urticae genotypes and again observed that the interaction between host genotype and the Wolbachia infection state significantly affected egg surface (χ 2 2=97.0092, p<0.0001) (Figure 4). Although we observed that Wolbachia infection only biased the sex ratio of Stt mites, we found that for both LonX and Stt mites, Wolbachia-infected females oviposited larger eggs compared to cured females (t-ratio=7.030, p<0.0001 and t-ratio=12.606, p<0.0001, respectively). However, on average, Wolbachia infection increased egg surface by ~11% in Stt mites and by only ~5% in LonX mites (Figure 4 and Table S5). In contrast, egg size was not significantly different within the Beis genotype (t-ratio=0.541, p=1.0000).

Egg size positively covaries with adult size in Tetranychus mites [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF], raising the question whether the Wolbachia-mediated increase in egg size also results in larger adults for the LonX and Stt genotypes. Using the projected body surface of teleiochrysalids (final molting stage before adulthood) as a proxy, our results show that the interaction between host genotype and the Wolbachia infection state significantly affected the adult size for both males and females (males: χ 2 1=14.5653, p=0.0001 and females: χ 2 1=11.4769, p=0.0007) (Figure 5). Wolbachia infection significantly increased male and female adult size within the Stt genotype (males: t-ratio=4.310, p=0.0002 and females: t-ratio=3.255, p=0.0058), but no significant differences were observed for LonX (males: t-ratio=0.998, p=1.0000 and females: t-ratio=1.542, p=0.5021) (Figure 5). For the Stt genotype, the projected body area of teleiochrysalid males and females were ~10% and ~5% larger in response to Wolbachia infection, respectively (Figure 5 and Table S6). This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1

Sex allocation distortion removes the threshold for Wolbachia invasion

We predicted the spread of Wolbachia in the three T. urticae nuclear genotypes and assessed the effect of sex allocation distortion on infection dynamics by adapting previously developed deterministic models for haplodiploids [START_REF] Vavre | Evidence for female mortality in Wolbachia -mediated cytoplasmic incompatibility in haplodiploid insects: Epidemiologic and evolutionary consequences[END_REF][START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF]. As observed for the LonX genotype, a single CI cross can lead to a mixture of FM-CI and MD-CI (Wybouw et al., 2022), an outcome that is now incorporated into the revised version of the model (Table S7 provides the full comparison between our and previous models).

For the Beis and LonX genotypes, simulations for Wolbachia invasion revealed that estimated fitness costs of Wolbachia infection result in unstable equilibria under which infection is lost after a number of host generations (i.e., the threshold infection frequency for Wolbachia invasion) (Figure 6). In particular, the high fecundity costs estimated in LonX leads to the highest threshold (~46% vs. ~16% in Beis; Figure 6). Above the threshold frequency, Wolbachia are predicted to spread the fastest in populations of the Beis nuclear genotype, in which total CI strength is the strongest (~100% in Beis, ~65% in LonX, and ~50% in Stt; Table S8). In contrast, despite similar fitness costs in Stt and Beis and weaker CI strength in the former, the model does not predict a threshold frequency for Wolbachia invasion in Stt (Figure 6). Additional simulations that revoke the effect of sex allocation distortion in the Stt genotype further show that the absence of an infection threshold is solely due to the Wolbachia-induced female-biased sex ratio (Figure S3). Indeed, without sex allocation distortion in the Stt genotype, only an initial infection frequency above ~24% would allow for Wolbachia invasion (Figure S3). In agreement with these results, there is no longer a threshold frequency for Wolbachia invasion in the Beis genotype after introducing the same level of sex allocation distortion as observed for Stt (Figure S3, the unstable equilibrium becomes ~0%). In contrast, the introduction of sex allocation distortion in the LonX genotype has only little effect (the invasion threshold decreases from ~46% to ~43%) (Figure S3). Finally, revoking CI in each of the three genotypes shows that Wolbachia should be purged from Beis and LonX but should still spread in Stt where sex allocation distortion compensates for the infection costs (Figure S3). Consistent with these observations, even though the Wolbachia variant induces weaker CI in Stt than in LonX (~50% vs. ~65% total CI strength), the infection is predicted to spread above the threshold frequency faster in Stt due to sex allocation distortion (Figure 6). Our model parametrization thus shows that, in some host genotypes, sex allocation distortion can strengthen the effect of CI and ensure the spread of Wolbachia independently of the initial infection frequencies. Together, these results corroborate previous theoretical predictions of the functional importance of sex allocation distortion for maternally inherited symbionts in haplodiploid hosts [START_REF] Egas | On the evolution of cytoplasmic incompatibility in haplodiploid species[END_REF].

Discussion

Maternally inherited symbionts commonly infect haplodiploid arthropods and spread through these host populations by inducing distinct reproductive phenotypes [START_REF] Weinert | The incidence of bacterial endosymbionts in terrestrial arthropods[END_REF][START_REF] Ma | Patterns and mechanisms in instances of endosymbiont-induced parthenogenesis[END_REF][START_REF] Shropshire | Symbiont-mediated cytoplasmic incompatibility: what have we learned in 50 years?[END_REF][START_REF] Kaur | Living in the endosymbiotic world of Wolbachia: A centennial review[END_REF]. A growing body of evidence indicates that sex allocation distortion could be a widespread, previously neglected, reproductive phenotype across divergent haplodiploid host -symbiont systems [START_REF] Vala | Genetic Conflicts over Sex Ratio: Mite-Endosymbiont Interactions[END_REF][START_REF] Wang | Intracellular symbionts drive sex ratio in the whitefly by facilitating fertilization and provisioning of B vitamins[END_REF][START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. Yet, several questions about the mechanisms that underpin sex allocation distortion are still outstanding, especially for Wolbachia.

The current results on the Stt nuclear genotype confirm that Wolbachia can cause higher female-biased sex ratios by regulating host sex allocation in Tetranychus mites [START_REF] Vala | Genetic Conflicts over Sex Ratio: Mite-Endosymbiont Interactions[END_REF]. Here, Wolbachia infection increases female production by ~11%, whereas [START_REF] Vala | Genetic Conflicts over Sex Ratio: Mite-Endosymbiont Interactions[END_REF] reports an increase of ~18% in the T. urticae C-strain. As for other symbiontmediated phenotypes, this difference in phenotypic strength can be attributed to genetic variation in Wolbachia and/or the arthropod host [START_REF] Turelli | Evolution of Incompatibility-inducing Microbes and Their Hosts[END_REF][START_REF] Engelstädter | The Ecology and Evolution of Microbes that Manipulate Host Reproduction[END_REF][START_REF] Beckmann | A single mutation weakens symbiont-induced reproductive manipulation through reductions in deubiquitylation efficiency[END_REF]Wybouw et al., 2022). The Wolbachia variant of the current study does not induce quantifiable sex allocation distortion in the Beis and LonX genotype, suggestive of strong host modulation by Tetranychus. These findings are consistent with a previous hypothesis that T. urticae evolves compensatory mechanisms that suppress Wolbachiainduced sex allocation distortion [START_REF] Vala | Genetic Conflicts over Sex Ratio: Mite-Endosymbiont Interactions[END_REF]. However, this hypothesis implies that sex allocation distortion is deleterious for Tetranychus hosts and that suppressor systems evolved in the Beis and LonX genotype to counteract the reproductive manipulations induced by Wolbachia.

As the focal Wolbachia variant also induces CI in incompatible matings (Wybouw et al., 2022), we show that Wolbachia are able to simultaneously induce CI and sex allocation distortion. These observations add to a growing body of literature showing that symbiont genomes often carry the genetic architecture of multiple reproductive manipulations [START_REF] Fujii | Transfection of Wolbachia in Lepidoptera: the feminizer of the adzuki bean borer Ostrinia scapulalis causes male killing in the Mediterranean flour moth Ephestia kuehniella[END_REF][START_REF] Kaur | Living in the endosymbiotic world of Wolbachia: A centennial review[END_REF][START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. Previous work uncovered that CI strength is also determined by host modulation within the T. urticae nuclear genotype reference panel (Wybouw et al., 2022). The penetrance of sex allocation distortion and CI appears uncoupled across the host genotypes. The Wolbachia variant only causes intermediate CI in Stt and LonX (~50% and ~65%, respectively), compared to complete CI in Beis (100%). We can therefore speculate that the host modifier systems act independently on the two Wolbachia-induced reproductive phenotypes. Host modulation of CI is largely manifested in infected Tetranychus males (Wybouw et al., 2022). As the reproductive physiology of haplodiploid females is a strong determinant of the sex ratio of their offspring [START_REF] Overmeer | Notes on the control of the sex ratio in populations of the twospotted spider mite, Tetranychus urticae Koch (Acarina: Tetranychidae)[END_REF][START_REF] Mitchell | The sex ratio of the spider mite Tetranychus urticae[END_REF][START_REF] Takafuji | Inheritance of sex ratio in the Kanzawa spider mite, Tetranychus kanzawai Kishida[END_REF][START_REF] Katlav | Constrained sex allocation after mating in a haplodiploid thrips species depends on maternal condition[END_REF] and sex allocation distortion appears independent of the infection state of males, it is likely that host modulation of sex allocation distortion occurs in females. Together, this further indicates that the host modifier systems of haplodiploids might have sex-specific phenotypic effects.

In this study, we also gathered strong evidence that Wolbachia-induced sex allocation distortion in Tetranychus mites is mediated by regulating egg size. For the Stt genotype, Wolbachia-infected females oviposit larger eggs compared to the Wolbachia-free sister line, increasing the probability of successful fertilization and female development [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF]Katlav, Cook, et al., 2021). We also noted a weaker, but significant, increase in egg size for the LonX genotype. Here, Wolbachia infection does not cause sex allocation distortion, likely because the egg size effect is too weak. As egg size also varied across the three T. urticae nuclear genotypes regardless of Wolbachia infection, our results identify egg size investment as a reproductive trait that is modulated by both the host and symbiont genotype. In Tetranychus females, a physiological trade-off controls resource allocation into egg number and size [START_REF] Macke | Mating Modifies Female Life History in a Haplodiploid Spider Mite[END_REF]. Here, Wolbachia infection influences egg number within the LonX nuclear background, but not within Stt. However, only a narrow 24 hour time interval of the oviposition period was monitored within our experimental set-up, raising the possibility that, conversely to [START_REF] Macke | Mating Modifies Female Life History in a Haplodiploid Spider Mite[END_REF], we missed the full trade-off effect of increased egg size on total egg number. Alternatively, this trade-off may be altered if Wolbachia infection causes an increase in available resources for ovipositing females. Many bacterial symbionts provide nutrients for their invertebrate host, including Wolbachia [START_REF] Hosokawa | Wolbachia as a bacteriocyte-associated nutritional mutualist[END_REF][START_REF] Kaur | Living in the endosymbiotic world of Wolbachia: A centennial review[END_REF]. Based on dietary supplementation experiments, symbiont-mediated provisioning of B vitamins is implicated in sex ratio distortion in haplodiploid whitefly species [START_REF] Wang | Intracellular symbionts drive sex ratio in the whitefly by facilitating fertilization and provisioning of B vitamins[END_REF]. In addition to egg size regulation, other Wolbachia-mediated changes in the female reproductive physiology could further contribute to sex allocation distortion. To increase the likelihood of successful egg fertilization, Wolbachia could facilitate the flow of sperm from the spermatheca to the ovary and the entry of sperm cells into the ovarian cavity [START_REF] Helle | Fertilization in the two-spotted spider mite (Tetranychus urticae: ACARI)[END_REF][START_REF] Helle | Spider mites. Their biology, natural enemies, and control[END_REF]. As sperm cells tend to accumulate in the dorsal region of the ovarian This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 cavity in Tetranychus mites, Wolbachia could also promote sperm-egg contact and subsequent egg fertilization by causing a more even sperm distribution within the cavity [START_REF] Feiertag-Koppen | Development of the female germ cells and process of internal fertilization in the two-spotted spider mite Tetranychus urticae koch (Acariformes : Tetranychidae)[END_REF][START_REF] Helle | Spider mites. Their biology, natural enemies, and control[END_REF].

Previous studies in haplodiploids uncovered a positive correlation between egg size, immature survival rate, and adult body size [START_REF] Macke | Sex allocation in haplodiploids is mediated by egg size: evidence in the spider mite Tetranychus urticae Koch[END_REF]Katlav, Cook, et al., 2021). Consistent with these studies, Cardinium-mediated increase in egg size is shown to be linked to higher juvenile survival rate and adult size in P. kellyanus [START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. Although Wolbachia-infected LonX-w and Stt-w females tended to have broods with higher immature survival rates than cured LonX-c and Stt-c females, we could not statistically demonstrate that Wolbachia infection positively affects this trait in these two genotypes. However, we found that Wolbachia infection increases male and female adult size within the Stt genotype, an effect that may lead to a higher reproductive success for infected adults [START_REF] Li | Does size matter? Fecundity and longevity of spider mites (Tetranychus urticae) in relation to mating and food availability[END_REF]. In Tetranychus, competition for mates among males can be intense, resulting in aggressive fighting behavior. Larger males have an advantage in these competitive interactions, and are more likely to fertilize virgin females under certain conditions [START_REF] Potter | Guarding, Aggressive Behavior, and Mating Success in Male Two-Spotted Spider Mites[END_REF][START_REF] Enders | The effect of male size and operational sex ratio on male mating success in the common spider mite, Tetranychus urticae Koch (Acari: Tetranychidae)[END_REF]. A higher fertilization success of Wolbachia-infected males may strengthen the effect of CI, further driving Wolbachia infection frequencies through host populations. In contrast, egg size difference does not persist into the adult stage in LonX mites, which is likely explained by the relatively weak egg size effect of Wolbachia infection in this nuclear genotype.

In haplodiploid Pezothrips insects, Cardinium symbionts also cause host sex allocation distortion [START_REF] Katlav | Common endosymbionts affect host fitness and sex allocation via egg size provisioning[END_REF]. The common strategy within the Wolbachia-Tetranychus and Cardinium-Pezothrips systems to distort sex allocation by regulating egg size raises the question of whether these symbionts share the same genetic basis for sex allocation distortion or evolved this reproductive phenotype independently. CI is also induced by both Cardinium and Wolbachia and is manifested by remarkably similar cytological defects [START_REF] Gebiola | Cytological analysis of cytoplasmic incompatibility induced by Cardinium suggests convergent evolution with its distant cousin Wolbachia[END_REF]. Yet, despite a nearly identical cellular phenotype, CI appears to be underpinned by different genetic architectures in these intracellular bacteria [START_REF] Gotoh | Cardinium symbionts cause cytoplasmic incompatibility in spider mites[END_REF][START_REF] Penz | Comparative Genomics Suggests an Independent Origin of Cytoplasmic Incompatibility in Cardinium hertigii (NA Moran[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachia-induced cytoplasmic incompatibility[END_REF][START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Beckmann | The Toxin-Antidote Model of Cytoplasmic Incompatibility: Genetics and Evolutionary Implications[END_REF][START_REF] Shropshire | Symbiont-mediated cytoplasmic incompatibility: what have we learned in 50 years?[END_REF]. Our results suggest that convergent mechanisms also underpin sex allocation distortion across divergent biological systems. To dissect the underlying mechanisms of sex allocation distortion, comparative genomics and transcriptomics data sets can be leveraged, for instance, the differentially expressed gene set of T. urticae embryos upon Wolbachia infection [START_REF] Bing | Transcriptome of Tetranychus urticae embryos reveals insights into Wolbachia -induced cytoplasmic incompatibility[END_REF].

Many haplodiploid arthropods, including thrips, whiteflies, and spider mites, are pathogen-carrying pests that threaten agricultural crop production [START_REF] Robertson | Virus-Like Particles and a Spider Mite Intimately Associated with a New Disease of Barley[END_REF][START_REF] Jones | Plant Viruses Transmitted by Thrips[END_REF][START_REF] Kitajima | An annotated list of ornamentals naturally found infected by Brevipalpus mite-transmitted viruses[END_REF][START_REF] Navas-Castillo | Emerging Virus Diseases Transmitted by Whiteflies[END_REF]. Pest control programs are being developed that rely on the driving ability of artificial Wolbachia infections that reduce pathogen transmission [START_REF] Ross | Evolutionary Ecology of Wolbachia Releases for Disease Control[END_REF][START_REF] Gong | Stable Introduction of Plant-Virus-Inhibiting Wolbachia into Planthoppers for Rice Protection[END_REF]. For successful population replacement, infected individuals must be released at a rate high enough to exceed threshold infection frequencies [START_REF] Ross | Evolutionary Ecology of Wolbachia Releases for Disease Control[END_REF]. Here, using a parametrized model, we show that sex allocation distortion could facilitate the spread of Wolbachia within haplodiploid pest populations by removing threshold infection frequencies and increasing invasion rates. Therefore, this reproductive phenotype may be instrumental to increase the efficacy of pest control measures. Our results underscore the importance of studying the impact and underlying mechanisms of natural and artificial Wolbachia infections on host sex allocation.

Wybouw N, Mortier F, Bonte D (2022). Interacting host modifier systems control Wolbachia -induced cytoplasmic incompatibility in a haplodiploid mite. Evolution Letters 6: 255-265.

Zélé F, Santos I, Matos M, Weill M, Vavre F, Magalhães S (2020). Endosymbiont diversity in natural populations of Tetranychus mites is rapidly lost under laboratory conditions. Heredity 124: 603-617.

Figure 3 .

 3 Figure 3. Effects of Wolbachia infection on fecundity depend on the host nuclear genotype. Daily oviposition from fertilized females for the infected and cured lines of the Beis, LonX, and Stt host genotype. Host genotypes are ordered according to decreasing CI strength.Statistical significance within each host genotype is outlined above: *, p < 0.05, whereas ns, not significantly different at the 5% level. Boxplots and replicates are color coded based on the infection state (see bottom right). Each boxplot represents the 25%, 50%, and 75% quantile.

Figure 4 .

 4 Figure 4. Wolbachia distort sex allocation in Tetranychus mites by regulating egg provisioning. Egg size from fertilized females for the infected and cured lines of the Beis, LonX, and Stt host nuclear genotype. Host genotypes are ordered according to decreasing CI strength. Statistical significance within each host genotype is outlined above: *, p < 0.05, whereas ns, not significantly different at the 5% level. Boxplots and replicates are color coded based on the infection state (see bottom right). Each boxplot represents the 25%, 50%, and 75% quantile.

Figure 5 .

 5 Figure 5. Wolbachia-induced size difference can persist into the adult stage of Tetranychus mites. The projected body surface of teleiochrysalids as a proxy for adult size for the infected and cured lines of the LonX and Stt host nuclear genotype (with males and females in panel A and B, respectively). Statistical significance within each host genotype is outlined above: *, p < 0.05, whereas ns, not significantly different at the 5% level. Boxplots and

Figure 6 .

 6 Figure 6. Distortion of host sex allocation is expected to facilitate Wolbachia invasion.Wolbachia infection frequencies throughout host generations for initial infection frequencies ranging from 0 to 1. Host genotypes are ordered according to decreasing CI strength. Dashed red lines represent the unstable polymorphic equilibrium, which corresponds to 'the threshold infection frequency for Wolbachia invasion'. Blue lines represent the stable polymorphic equilibrium, which is reached when the initial infection frequency is above the unstable equilibrium. Our parametrized model does not predict a threshold infection frequency in Stt despite a lower oviposition estimate upon Wolbachia infection. Additional simulations that add/remove sex allocation distortion in Beis, LonX, and Stt further support the contribution of sex allocation distortion to Wolbachia invasion (FigureS3).

. Wolbachia induce two distinct reproductive phenotypes in Tetranychus mites.

  As Tetranychus mites have a haplodiploid mode of reproduction, unfertilized eggs develop into haploid males, while fertilized eggs generate diploid females (control, top left). Wolbachiamediated sex allocation distortion facilitates successful egg fertilization and subsequent female development, biasing population sex ratios towards females. In CI crosses, fertilized eggs suffer from higher mortality rates (Female Mortality CI, FM-CI), or a proportion of fertilized eggs reach adulthood as adult males (Male Development CI, MD-CI). A single CI cross can lead to a mixture of FM-CI and MD-CI. Here, CI strength varies across the illustrative CI crosses.

	Figure 2.

Wolbachia-mediated sex allocation distortion is contingent on the host nuclear genotype. Panel A:

  Sex ratio from compatible within-line crosses for the infected and cured lines of the Beis, LonX, and Stt host genotype. Sex ratio was calculated as the proportion of females among the progeny that successfully reached adulthood. Host genotypes are ordered according to decreasing CI strength. Statistical significance within each host genotype is outlined above: *, p < 0.05, while ns, not significantly different at the 5% level. Panel B: Sex ratio from compatible within-line crosses for the uninfected, infected, and cured line of the Stt host genotype. Identical superscripts outline non-significant differences at the 5% level. For both panels, boxplots and replicates are color coded based on the infection state (see bottom middle). Each boxplot represents the 25%, 50%, and 75% quantile.

This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1

Acknowledgements

We thank Masahiko Tanahashi for assisting with the Natsumushi analyses, Sara Magalhães and Fabrice Vavre for useful discussions, Martijn Egas for confirming the error in Egas et al., This is a post-peer-review, precopyedit version of an article published in Heredity. The final authenticated version is available online at: https://doi.org/10.1038/s41437-023-00638-1 2002, and Guillaume Martin for his help with identifying the equilibria of the mathematical model. We also thank Bouwe Cattrysse for generating exploratory pilot data for the LonX-c and LonX-w lines. NW was supported by a BOF post-doctoral fellowship (Ghent University, 01P03420) and by a Research Foundation -Flanders (FWO) Research Grant (1513719N). This work was further supported by the FWO Research Network EVENET. This is contribution ISEM-2023-125 of the Institute of Evolutionary Science of Montpellier (ISEM).

Data archiving

All raw data and R scripts are available at the Dryad repository.

Author contribution statement

NW conceived and designed the experiments. NW, EVR, and JZ performed the experiments. FZ extended the model and performed the simulations. NW, FZ, and DB analyzed the data. NW wrote the manuscript with input from FZ and DB. All authors read and approved the final version of the manuscript.

Conflict of Interest

The authors declare no conflicts of interest.


Daniel Alpay

Mamadou Mboup
email: mamadou.mboup@mi.parisdescartes.fr

A natural transfer function space for linear discrete time-invariant and scale-invariant systems

In a previous work, we have defined the scale shift for a discrete-time signal and introduced a family of linear scale-invariant systems in connection with characterautomorphic Hardy spaces. In this paper, we prove a Beurling-Lax theorem for such Hardy spaces of order 2. We also study an interpolation problem in these spaces, as a first step towards a finite dimensional implementation of a scale invariant system. Our approach uses a characterization of character-automorphic Hardy spaces of order 2 in terms of classical de Branges Rovnyak spaces.

I. INTRODUCTION

The self-similarity property is widely studied in the literature in the framework of stochastic process theory [START_REF] Mboup | Discrete-time models for statistically self-similar signals[END_REF], [START_REF] Leland | On the selfsimilar nature of Ethernet traffic (extended version)[END_REF], [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF], and in the framework of systems theory [START_REF] Mboup | On the structure of self-similar systems: A Hilbert space approach[END_REF], [START_REF] Yazıcı | A class of second-order stationary self-similar processes for 1/f phenomena[END_REF], [START_REF] Nuzman | Reproducing kernel Hilbert space methods for wide-sense self-similar processes[END_REF]. In stochastic process theory the property is seen as a weighted form of stationarity in scale while in the systems theory approach, it is interpreted as a scale invariance. The scale shift, defined for a signal x(t) by the operator α → x(αt) thus plays a central rôle in the definition of self-similarity. Though simple and straightforward in the continuous-time domain, this operator is not well defined for discrete-time signal. In this paper, we use the definition given in [START_REF] Mboup | A character-automorphic hardy spaces approach to discrete-time scale-invariant systems[END_REF]. Therein, a family of linear discrete both timeand scale-invariant systems is introduced in connection with character-automorphic Hardy spaces. In this paper, we prove a Beurling-Lax theorem for such Hardy spaces of order 2. We also study an interpolation problem in these spaces, as a first step towards a finite dimensional implementation of a scale invariant system. Our approach uses a characterization of character-automorphic Hardy spaces of order 2 in terms of classical de Branges Rovnyak spaces [START_REF] Alpay | A characterization of Schur multipliers between character-automorphic hardy spaces[END_REF].

A. Scale shift for discrete-time signals

Let f ∈ L 1 (R + ) (that is, a continuous time signal), with Laplace transform F (s), ℜ(s) 0.
As it is well known, for every α = 1/β > 0, the Laplace transform of f (βt) is √ αF (αs). Therefore, time scaling has the same form both in the time and frequency domains. This remark is the starting point to define the scaling operator for discrete-time signals.

D = {z ∈ C : |z| < 1}.
Then, the scale shift

S α : s → S α (s) = αs, α > 0
translates in the unit disc, into the hyperbolic transformation [START_REF] Mboup | A character-automorphic hardy spaces approach to discrete-time scale-invariant systems[END_REF]

γ {α} = G θ • S α • G -1 θ . (1)
Any transformation of this form maps the unit open disc (resp. the unit circle) into itself. Conversely, the following lemma is true. Lemma 1.1: For each hyperbolic transformation

γ(z) = γ 1 z + γ 2 γ 2 z + γ 1 , (2)
there exist α γ > 0, θ γ with |θ γ | < π 2 and ξ γ such that

e iξγ γ(z) = (G θγ • S αγ • G -1 θγ )(e iξγ z). (3)
In particular, α γ is given by the multiplier of the transformation Proof: We assume that γ is normalized such that |γ 1 | 2 -|γ 2 | 2 = 1. Since γ is hyperbolic, we have the relation [START_REF] Ford | Automorphic functions[END_REF]

γ(z) -ξ 1 γ(z) -ξ 2 = K z -ξ 1 z -ξ 2 (4)
where

ξ 1 = √ [ℜ(γ1)] 2 -1+iℑ(γ1) γ2
= λγ γ2 and ξ 2 = -λγ γ2 are the two fixed points of γ. The positive constant K is called the multiplier of the transformation [START_REF] Ford | Automorphic functions[END_REF] and is given by

K + 1 K = 4[ℜ(γ 1 )] 2 -2. Noting that |ξ 1 | = |ξ 2 | = 1, one may rearrange (4) to obtain λ γ -λ γ e iξγ γ(z) 1 + e iξγ γ(z) = K λ γ -λ γ e iξγ z 1 + e iξγ z , (5)
where e iξγ = The set of all linear transformations as in (2) forms a group that we denote by Γ. The lemma then shows that the action of Γ on D is equivalent to the scale operator on C + . Therefore, we define below the discrete-time frequency domain scale shift by the action of the (hyperbolic) group of automorphisms of D. Given a discrete-time signal {x n } n≥0 , consider its Z transform X(z) = ∞ n=0 x n z n , which we assume convergent in a neighborhood of the origin. The scale α = α γ shift of the sequence {x n } n≥0 is the sequence {x n (γ)} n≥0 defined via the equation

X γ (z) = 1 γ 2 z + γ 1 X (γ(z)) = n≥0 x n (γ)z n . (6)
It is useful to note that this operator also makes sense for vector-valued functions.

B. Scale-invariant systems

A wide class of causal discrete time-invariant linear systems can be given in terms of convolution in the form

y n = n m=0 h n-m x m , n = 0, 1, . . . , (7)
where {h n } is the impulse response and where the input sequence {x m } and output sequence {y m } are requested to belong to some pre-assigned sequences spaces. The Z transform of the sequence {h n }, that is H(z) = ∞ n=0 z n h n , is called the transfer function of the system, and there are deep relationships between properties of H(z) and of the system; see [START_REF] Alpay | of Panoramas et Synthèses[END_REF] for a survey. In particular, it is well known that if the system is asymptotically stable, then H(z) belongs to the classical Hardy space of order 2.

In this paper, we are interested in linear discrete-time systems which, in addition to the time invariance, are also invariant under a scale shift. Scale-invariance is defined [START_REF] Mboup | A character-automorphic hardy spaces approach to discrete-time scale-invariant systems[END_REF] similarly to time-invariance: a scale shift in the input sequence induces the same scale shift on the corresponding output sequence. In terms of the Z transforms, this would directly mean that for all γ ∈ Γ,

Y γ (z) = H(γ(z))X γ (z) = H(z)X γ (z). (8)
Therefore, scale-invariance implies that the transfer function of the system be Γ-periodic. Now a function f satisfying f • γ = f , for all γ ∈ Γ is said to be automorphic with respect to Γ. This makes sense only for discrete groups (see [START_REF] Ford | Automorphic functions[END_REF]).

In the following, we will be interested in the characterautomorphic Hardy spaces of order 2. These are the natural transfer function spaces for the LTI and scale-invariant systems. In a previous study, see [START_REF] Alpay | A characterization of Schur multipliers between character-automorphic hardy spaces[END_REF], we have given a characterization of these spaces in terms of associated classical de Branges Rovnyak spaces. In section II, we use this approach to prove a Beurling-Lax theorem and we study interpolation in these Hardy spaces in section III. Leech's theorem and the characterization of de Branges Rovnyak spaces given in [11, Theorem 3.1.2, p. 85] play an important role in the arguments.

In the remaining of the paper, the complex conjugation of is denoted par * and no longer by .

II. A NATURAL TRANSFER FUNCTION SPACE

A. Definitions

Now on, we discretise the scale axis and consider that Γ is a Fuchsian group of Widom type (with no elliptic element) [START_REF] Samokhin | Some classical problems in the theory of analytic functions in domains of Parreau-Widom type[END_REF]. We denote by z its uniformizing map and by Γ its dual group, i.e. the group of unimodular characters. Recall that a character α is a function defined on Γ and satisfying:

|α(γ)| = 1 and α(γ • ϕ) = α(γ)α(ϕ), ∀γ, ϕ ∈ Γ. A function f satisfying f • γ = α(γ)f, ∀γ ∈ Γ for α ∈ Γ,
is called character-automorphic with respect to Γ. Given a character α of Γ, the character-automorphic Hardy space H α 2 (D) of order 2 is the space of character-automorphic functions which belong to the classical Hardy space H 2 (D). Its reproducing kernel has been characterized in [13, Lemma 4.4.2 p. 387] as follows:

k α (z, ω) = c(α) k αµ (z,0) b(z) k α (ω, 0) * -k αµ (ω,0) b(ω) * k α (z, 0) z(z) -z(ω) * (9) where c(α) = z(0)b(0)
k αµ (0,0) > 0. In ( 9), b is the Green's function of Γ, and the character associated to the Green's function is denoted by µ. Formula (9) expresses that the kernel is structured, and of the form of the kernels studied in the papers [START_REF] Alpay | On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains[END_REF], [START_REF] Alpay | On a new class of structured reproducing kernel spaces[END_REF]. It depends on a C 1×2 -valued function of one variable. Using (9) we proved in [START_REF] Alpay | A characterization of Schur multipliers between character-automorphic hardy spaces[END_REF] that there exists a Schur function S α , associated to the de Branges space H(S α ), such that

H α 2 (D) = F (z) = A α (z) 1 -iz(z) f (σ(z)) ; f ∈ H(S α ) (10) where A α (z) = c(α) k αµ (z,0) b(z) + ik α (z, 0) and σ(z) = 1+iz(z)
1-iz(z) , and with the norm

F H α 2 (D) = f H(Sα)
. We close this subsection with the Definition 2.1: A causal linear time-invariant system is called scale-invariant with respect to Γ, if its transfer function is an element of H α 2 (D) for some character α. Note that such a system is not rational, unless Γ is a finite group.

B. The shift operator in H α

2 (D) Set p ∈ D and given a function f analytic in D, consider the operators

(R p f )(λ) = f (λ) -f (p) λ -p .
These operators R p satisfy the resolvent equation

R p -R q = (p -q)R p R q .
Let m(z) = Aα(z) 1-iz(z) . The isomorphism

F (z) = m(z)f (σ(z)) (11)
between the de Branges space H(S α ) and the characterautomorphic Hardy space allows one to define the following operators:

R p F = m(z)(R p f )(σ(z)).
As we may directly check, these operators R p also satisfy a resolvent equation. Therefore, they can be written in the form

R p = (T -p) -1 .
Here T is not an operator in general. It is a linear relation and is given by

(TF )(z) = σ(z)F (z) + m(z)c F ,
where c F is such that

σ(z)F (z) + m(z)c F ∈ H α 2 (D). C. A Beurling theorem in H α 2 (D)
The classical Beurling theorem (see for instance [START_REF] Rudin | Analyse réelle et complexe[END_REF]Théorème 17.21 p. 330]) gives a characterization of the closed subspaces M of the Hardy space H 2 (D) of the unit disk D: Any such subspace is of the form M = jH 2 (D), where j is an inner function (the case of vector-valued functions was first considered by Lax; see [START_REF] Lax | Scattering theory[END_REF] for a discussion and references). The orthogonal complement of M is the reproducing kernel Hilbert space with reproducing kernel K j (z, w) = (1 -j(z)j(w) * )/(1 -zw * ). If one replaces j by a Schur function s, that is, by a function analytic and contractive in D, the kernel k s (z, w) is still positive in D. Its associated reproducing kernel Hilbert space was denoted in the preceding subsection by H(S ). These spaces are called de Branges Rovnyak spaces, and originate with the work [START_REF] De Branges | Square summable power series[END_REF]. When allowing S to be vector-valued, they have been fully characterized in [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]Theorem 3.1.2]. They are contractively included, but in general not isometrically included, in H 2 (D).

Theorem 2.2: A Hilbert space M is contractively included in H α 2 (D), and invariant under R p , and satisfies the inequality

R 0 F 2 M ≤ F 2 M -|F (0)| 2 (12)
if, and only if, its reproducing kernel is of the form

A α (z) √ 2 1 -R(σ(z))R(σ(w)) * -i(z(z) -z(w) * ) A α (w) * √ 2 ,
where R is a vector-valued Schur function such that

S α = RR 1 , (13)
with R 1 also a vector-valued Schur function.

Remark 1: The inequality( 12) is automatically satisfied if M is isometrically included in H α 2 (D). Proof: of 2.2 Associated to the space M, there is, by the isomorphism [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF], a Hilbert space M α which is R p -invariant and satisfying

R 0 f 2 Mα ≤ f 2 Mα -|f (0)| 2 . (14)
Using [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF]Theorem 3.1.2], we see that the reproducing kernel of M α is of the form

1 -R(λ)R(p) * 1 -λp * ,
where R is a vector-valued Schur function. Since M α is contractively included in H(S α ), the kernel

R(λ)R(p) * -S α (λ)S α (p) * 1 -λp * ,
is positive. We therefore get the factorization (13) by using the Leech theorem. See [START_REF] Rosenblum | A corona theorem for countably many functions[END_REF]Theorem 2,p. 134] and [START_REF] Rosenblum | Hardy classes and operator theory[END_REF]Example 1,p. 107]. We conclude by using the isomorphism [START_REF] Alpay | Schur functions, operator colligations, and reproducing kernel Pontryagin spaces[END_REF] and the fact that (see equation ( 10)) the reproducing kernel of the space

H α 2 (D) is k α (z, w) = A α (z) 1 -iz(z) 1 -S α (σ(z))S α (σ(w)) * 1 -σ(z)σ(w) * A α (w) * 1 + iz(w) * = A α (z) √ 2 1 -S α (σ(z))S α (σ(w)) * -i(z(z) -z(w) * ) A α (w) * √ 2 .
Reversing these different arguments allows one to establish the converse.

III. INTERPOLATION

As we already mention, the elements of the characterautomorphic Hardy space H α 2 (D) are not rational functions unless we consider a finite number of possible scale shifts. Since the corresponding systems are of infinite dimension, a finite dimensional approximation step is necessary before their implementation. This is the motivation of the interpolation problem studied in this section.

To proceed, denote by

F = {z ∈ D : |γ ′ (z)| < 1 for all γ ∈ Γ, γ = id} (15)
the normal fundamental domain of Γ with respect to 0: there is no transformation in Γ, which sends one point of F into another point of F. So, we consider the following interpolation problem:

Given N complex numbers F i and N points z i ∈ F ∩ D, describe the set of all functions F ∈ H α 2 (D) with F H α 2 (D) ≤ 1 satisfying F (z i ) = F i , i = 1, . . . N, (16)
Note that this problem is different from the interpolation problems considered by Abrahamse in [START_REF] Abrahamse | The Pick interpolation theorem for finitely connected domains[END_REF] and by Kupin and Yuditskii in [START_REF] Kupin | Analogs of the Nehari and Sarason theorems for character-automorphic functions and some related questions[END_REF]. These studies were interested in finding multipliers having given values at prescibed points while here we have the constraint that F must belong to H α 2 (D). The function

P α = (1 -S α ) (1 + S α ) (17)
is analytic and has positive real part in D. By the Herglotz representation theorem, we can write:

P α = ic α + 2π 0 e it + λ e it -λ dσ α (t), with c α ∈ R and dσ α is a postive measure [0, 2π). Defining Q α (λ) △ = √ 2 (1 + S α (λ)) = 1 + P α (λ) √ 2
where the second equality follows directly from ( 17), we see that can also write

P α (λ) + P α (p) * 1 -λp * = Q α (λ) 1 -S α (λ)S α (p) * 1 -λp * Q α (λ) * .
(18) Now this equation [START_REF] De Branges | Square summable power series[END_REF] implies that the operator of multiplication by Q α (λ) is a unitary transformation from the reproducing kernel space L(P α ), with kernel

P α (λ) + P α (p) * 1 -λp * ,
with h ∈ H 2 (dσ α ).

The interpolation problem in the character-automorphic Hardy space then reduces to an interpolation problem in L(P α ), or more precisely, to an orthogonal projection in We thus deduce the description of all x of the form [START_REF] Rosenblum | A corona theorem for countably many functions[END_REF], and hence a description of the functions F by formula [START_REF] Rosenblum | Hardy classes and operator theory[END_REF].

λγ γ2 .

 γ2 Dividing both sides of this equality by |λ γ |, we get (3) by setting e iθγ = λγ |λγ | and α γ = K.

H 2 ( 1 : 0 e

 210 D, dσ α ): Problem Let λ ℓ = σ(z ℓ ) andx ℓ = 1z(z ℓ ) A α (z ℓ ) √ 2F ℓ 1 + S α (λ ℓ )) , ℓ = 1, . . . N. Find all h ∈ H 2 (D, dσ α ) such that 2π it h(t)dσ α (t) e it -λ ℓ = x ℓ , ℓ = 1, . . . N.Now this is a classical Hilbert space problem: it admits a solution with minimum norm, corresponding to a function h min of the formh min (t) = N ℓ=1 c ℓ e it -λ ℓ ,and any other solution has the formh min + h, h ⊥ h min .

  λ, p ∈ D to the space H(S α ). Moreover L(P α ) is the set of functions of the form belongs to the closure (subsequently denoted H 2 (D, dσ α )) of the set of functions 1/(1 -e it w * ) (|w| < 1) in L 2 (dσ α ). We therefore have the following: Proposition 3.1: F ∈ H α 2 (D) if, and only if,

	where h F (z) =	x(λ) = A α (z) 1 -iz(z) 1 + S α (σ(z)) 2π 0 e it h(t)dσ α (t) e it -λ 2 0 √ 2π e	,	(19)

it h(t)dσ α (t) e it -σ(z)