Weak antagonistic fitness effects can maintain an inversion polymorphism

The study of chromosomal inversion polymorphisms has received much recent attention, particularly in cases where inversions have drastic effects on phenotypes and fitness (e.g. lethality of homozygotes). Less attention has been paid to the question of the maintenance of inversion polymorphisms that show only weak effects. Here, we study the maintenance of such an inversion polymorphism that links 250 genes on chromosome Tgu11 in the zebra finch (Taeniopygia guttata). Based on data from over 6000 captive birds, we estimated the effects of this inversion on a wide range of fitness‐related traits. We found that, compared with the ancestral allele A, the inverted allele D had small additive beneficial effects on male siring success and on female fecundity. These fitness‐enhancing effects may explain the initial spread of the derived D allele (allele frequency 53%). However, individuals that were homozygous for D had a slightly lower survival rate, which may explain why the D allele has not spread to fixation. We used individual‐based simulations to examine how an inversion polymorphism with such antagonistic fitness effects behaves over time. Our results indicate that polymorphisms become stabilized at an intermediate allele frequency if the inversion links an additively beneficial allele of small effect size to a recessive weakly deleterious mutation, overall resulting in weak net heterosis. Importantly, this conclusion remains valid over a wide range of selection coefficients against the homozygous DD (up to lethality), suggesting that the conditions needed to maintain the polymorphism may frequently be met. However, the simulations also suggest that in our zebra finch populations, the estimated recessive deleterious effect of the D allele (on survival in captivity) is not quite large enough to prevent fixation of the D allele in the long run. Estimates of fitness effects from free‐living populations are needed to validate these results.


| INTRODUC TI ON
Chromosomal inversions are widespread and are increasingly found to be important in explaining phenotypic variation, adaptation and speciation (Faria et al., 2019;Fuller et al., 2019;Hoffmann & Rieseberg, 2008;Mérot, Oomen, et al., 2020). Due to suppressed recombination between the inverted and the ancestral variants, the inversion can physically link hundreds of loci into one supergene (Kirkpatrick, 2010). Once an inversion has arisen in a population, the evolutionary fate of the inversion will partly depend on chance (with genetic drift being larger in small populations) and partly on the inversion's effects on fitness (Gutiérrez-Valencia et al., 2021;Llaurens et al., 2017). Inversions may get fixed or lost over time as the conditions required to maintain them in a polymorphic state are not trivial (Faria et al., 2019;Gutiérrez-Valencia et al., 2021), yet many species have inversions that segregate at intermediate allele frequencies within or between populations (Harringmeyer & Hoekstra, 2022;Huang et al., 2022;Wellenreuther & Bernatchez, 2018).
Inversions are often underdominant because heterozygous individuals have problems during meiotic cell division. In heterozygous carriers of the inversion, cross-overs in the inverted region can lead to unbalanced and hence unviable gametes. Yet, the frequency of such detrimental cross-overs varies considerably between taxa and also depends on the size of the inversion and its position relative to the centromere (for details see Anton et al., 2005;Fuller et al., 2019;Knief et al., 2016;Roberts, 1967;Torgasheva & Borodin, 2010). In some cases, recombination in the inverted region is effectively suppressed, such that offspring mortality remains minimal or limited to very large inversions that cover more than half of a chromosome (Anton et al., 2005;Fuller et al., 2019;Knief et al., 2016).
While such selection against heterozygotes (underdominance) would promote the loss of the rarer allele, there is also a range of factors that can promote the maintenance of a polymorphism, either individually or jointly. (1) Overdominance, in which heterozygous individuals have higher fitness than homozygous carriers. As an extreme example, in Drosophila tropicalis only individuals that are heterozygous for an inversion on chromosome 2 can survive to reach adulthood (Dobzhansky & Pavlovsky, 1955). (2) Antagonistic pleiotropy, in which the genes linked within an inversion show opposing effects on different fitness-related traits (often resulting in net heterosis). For instance, in the seaweed fly Coelopa frigida, genes inside an inversion on chromosome I influence female fecundity and offspring survival in opposite directions (Betran et al., 1998;Mérot, Llaurens, et al., 2020). (3) Frequency-dependent selection, in which fitness is a function of the frequency of the inversion alleles. In the ruff Philomachus pugnax, negative frequency-dependent selection and intralocus sexual conflict contribute to the maintenance of the Feader inversion allele, which controls one of three male reproductive phenotypes (Giraldo-Deck et al., 2022;Küpper et al., 2015;Lamichhaney et al., 2015). (4) Disassortative mating between inversion genotypes in the white-throated sparrow Zonotrichia albicollis (Tuttle, 2003;Tuttle et al., 2016) and in the butterfly Heliconius numata (Chouteau et al., 2017). (5) Spatial or local adaptation, typically manifested as changes in the frequency of the inverted allele along an environmental gradient. Such cases have been described, for example, in the yellow monkeyflower Mimulus guttatus (Lee et al., 2016;Lowry & Willis, 2010), in the fruit fly Drosophila melanogaster (Durmaz et al., 2018;Kennington et al., 2007), in the quail Coturnix coturnix (Sanchez-Donoso et al., 2022) and in the deer mouse Peromyscus maniculatus (Hager et al., 2022). (6) Interchromosomal incompatibilities between loci (epistasis or Bateson-Dobzhansky-Muller incompatibility), in which multiple chromosomal inversions may co-adapt in different populations leading to fixation of different inversions in different populations, while hybrids suffer from incompatibilities of allelic combinations. Although such cases are difficult to detect due to the complexity of multilocus interactions, the nonrandom co-occurrence of two inversions on different chromosomes in Drosophila melanogaster can be considered an example (Singh & Das, 1991). The study reported a significant excess of individuals that were dual homozygous for the ancestral types or dual heterozygous for both inversions.
Our understanding of the evolution of inversion polymorphisms is limited and biased to systems that have large phenotypic effects (Rockman, 2012; also see Pracana et al., 2017;Lindtke et al., 2017).
However, often-and in particular for inversions that have no obvious phenotypic effects-the mechanism(s) behind the maintenance of the inversion polymorphism remain unclear (Faria et al., 2019;Wellenreuther & Bernatchez, 2018). For instance, in the deer mouse Peromyscus maniculatus, an inversion on chromosome 15 was associated with forest and prairie ecotypes (Hager et al., 2022), but out of another 20 large chromosomal inversions only four had apparent effects on phenotypes and fitness (Harringmeyer & Hoekstra, 2022).
Zebra finches have at least four large chromosomal inversions that all segregate at around 50% allele frequencies, both in the wild and in captivity (Christidis, 1986;Itoh et al., 2011;Knief et al., 2016).
In contrast to the iconic examples listed above, these four large inversions, capturing more than 250 genes each, have no striking effects on phenotypes and no immediately apparent fitness effects (e.g. no lethality of homozygotes of the derived allele). However, a previous study showed that about half of eight morphological traits were affected by each of the four inversions (Knief et al., 2016). Hence, many morphological traits may be affected when the allelic effects of more than 250 genes are combined into a single haplotype. This finding supports the notion that morphological traits have a highly polygenic architecture (Knief et al., 2016). Because fitness is thought to be one of the most polygenic traits, and a particularly large target to be influenced by de novo mutations (Veltman & Brunner, 2012), we expect that inversions will often capture multiple loci with effects on fitness-related traits.
Among the four chromosomal inversions of the zebra finch, only the largest one, located on the sex chromosome TguZ, appears to be maintained by a moderately strong overdominance effect (Kim et al., 2017;Knief et al., 2017. The mechanism maintaining the other three inversion polymorphisms remains unclear (Knief et al., 2016). Here, we aim to elucidate the mechanism underlying the maintenance of the inversion on chromosome Tgu11. This inversion segregates independently from the sex chromosomal inversion on TguZ (r 2 = 0.002, p = .1; Knief et al., 2016); it captures 250 genes and spans 57% of the assembled chromosome (12 Mb; Knief et al., 2016).
Previous work found no sign of underdominance due to problems in meiosis, but rather suggested that the derived (inverted) type had a weak beneficial effect on fitness in captive zebra finches (Knief et al., 2016). However, the selective forces that would maintain the polymorphism remain unknown.
Here, we first quantified the fitness effects of the Tgu11 inversion, using data on individual survival and reproductive performance from four captive populations of zebra finches, based on up to 14 years of breeding with more than 6000 captive birds. Second, we ran individual-based simulations to investigate how the obtained fitness estimates of alleles would affect the short-term evolution (i.e. equilibrium frequency) of the inversion polymorphism. For this, we simulated populations containing the inversion and used the estimated effect sizes on fitness to assess frequency changes in the inversion over time. We then explored the conditions under which maintenance of the polymorphism is achieved by extending the simulations to a wider range of selection differentials against individuals with potentially detrimental allele combinations.

| Tag SNP genotyping
We here refer to the ancestral genotype as allele A and to the derived, inverted genotype as allele D (in contrast to Knief et al., 2016, who labelled the ancestral genotype as allele B because its allele frequency was below 50% and the derived genotype as allele A).
We selected one tag SNP (WZF00031805, see Table S2 for details) from Knief et al. (2016) based on the highest linkage disequilibrium (LD) with the inversion haplotype (Pearson correlation coefficient r = 0.998, p < .0001 based on 948 individuals from the wild) and a missing call rate of <0.01 (due to a few samples of apparent low quality). The error rate of genotyping was 0.03, based on 326 zebra finches that were genotyped twice for the selected tag SNP in Knief et al. (2016). Errors were mostly due to failing to assign the heterozygote.
Inversion types of 4951 birds were taken from Knief et al. (2016).
An additional 1955 birds were newly genotyped via the selected tag SNP using the Roche LightCycler Instrument following the manufacturer's guide. To study the effects of the inversion genotype on offspring survival, we further genotyped 3022 dead embryos and 1522 dead nestlings. For a subset of the genotyped individuals, we collected reproductive performance and survival data (for details see below). Genotypes were called by the default parameters on the Roche LightCycler Software. Eight out of 11,458 cases (total number of genotyped individuals, including adults) failed the inheritance check and were removed from analysis. These failed cases (0.07%) were either embryos (N = 7) or nestlings that had died at a young age (N = 1), and the inheritance errors were due to parthenogenesis (4 cases), to incorrect parentage assignment (3 cases) or to falsely assigning an individual as homozygous (1 case). The allele frequency of the tag SNP in wild birds and among the captive populations is shown in Table S1.

| Fecundity, fertility and viability measurements
Data on reproductive performance, lifespan and nestling body mass at 8 days of age were taken from  and . Reproductive performance traits had been measured either (1) in 'cages' containing a single breeding pair or (2) in 'aviaries' with a group of males and females (typically 12 birds with balanced sex ratio, range: 10-42 birds and 40-60% of males) where breeding pairs were formed freely. During the breeding seasons, nests were checked daily to record hatching and death dates. Figure S1 shows the estimated age at death of the offspring (in days of incubation for embryos and in days posthatching for hatchlings). Embryonic age at death (in days) was estimated by one observer (K. Martin) based on the stage of embryo development. Details on rearing conditions, timing of the breeding seasons, and all measured reproductive traits are described in .
In brief, we studied four female reproductive performance traits: (1) clutch size in cages (N = 562 females), (2) clutch size in aviaries (N = 703 females), (3) the total number of eggs laid over the entire breeding season in aviaries where offspring rearing was not allowed and eggs were replaced by plastic eggs (N = 285 females) and (4) the number of independent young a female produced (genetic mother, determined using genotyping with microsatellite markers) within a breeding season in aviaries where offspring rearing was allowed (N = 438 females). For males, we studied (5) egg fertility in cages; that is, whether an egg laid in a cage was fertilized (N = 504 males), (6) egg siring success in aviaries, that is whether an egg laid by the male's social partner was fertilized by him (N = 512 males; determined using genotyping with microsatellite markers), (7) the total number of eggs sired in an aviary, including both within-and extrapair (N = 724 males), and (8) the total number of independent young sired in an aviary (N = 432 males). Overall, we monitored at least one of these reproductive performance traits for 1182 females and for 984 males.
We also estimated the effect of the Tgu11 inversion on individual (offspring) survival. Here, we included (9) the effect of the genotype of the genetic mother on embryo hatching success (N = 940 females), because  found that the identity of the genetic mother significantly predicted embryo survival.

| Statistical analyses
We used the function 'cor.test' from the 'stats' package in R V3.6.1 (R Core Team) to calculate the Pearson's product-moment correlation of the number of copies of the derived inversion that all genotyped captive birds carried between the chromosomes TguZ and Tgu11 We used a weighted 'lm' function from the 'stats' package in R V3.6.1 to study how the derived D allele frequency had changed over time in our four captive populations. For this, we calculated for each population and each year the D allele frequency among newly produced recruits that survived to at least 100 days of age. This frequency was treated as the response variable, and each data point was weighted by the number of recruits. Then, we estimated the effect of year as a linear covariate, while fitting population ID as a fixed effect to control for between-population differences in allele frequency due to founder effects.
We fitted mixed-effect models using the function 'lmer' in the package 'lme4' (Bates et al., 2015) in R V3.6.1 to estimate effects of inversion genotypes on fitness. We adapted the model structures used in  on reproductive performance, offspring survival and lifespan (Tables S3 and S4 show   sample sizes and all fixed effects, while Table S5 lists all random effects). Note that we square-root-transformed the total number of eggs and the total number of independent offspring produced by a female, the total number of eggs sired by a male, the total number of independent offspring produced by a male, and lifespan, to approach normality. Binomial traits of fertility and egg survival were modelled at the level of single eggs (coded as 0 or 1), yet we used a Gaussian error structure in all mixed-effect models Schielzeth et al., 2020) to obtain standardized effect sizes after Z-scaling each of the 13 fitness traits (Nakagawa & Cuthill, 2007;Nakagawa & Schielzeth, 2013). We fitted the inversion genotype as a fixed effect with three levels (AA: homozygous ancestral, AD: heterozygous and DD: homozygous derived) and showed the result relative to the AA ancestral baseline (Table S3).
We controlled for the following Z-scaled covariates in all models: inbreeding (i.e. pedigree-based inbreeding coefficient F), age at the start of reproduction (i.e. age of the male or female when the female laid the first egg; in days) or age at measurement (95% of nestlings were weighed at 8 days of age, but some at Days 7, 9 or 10), the number of days the focal bird participated in a breeding season and the sex ratio (i.e. the proportion of males in an aviary). Whenever applicable (see , we also controlled for laying order (egg within a clutch), hatching order (nestling within a brood), clutch order (within a breeding season), whether an egg was laid while nestlings from a previous clutch were still present (yes/no), pair formation type (i.e. forced pairs in cages or free-choice in aviaries), cross-fostering regime (i.e. no cross-fostering, cross-fostered to parents from the same population, domesticated nestlings being cross-fostered to wild-derived parents, and wild-derived nestlings being cross-fostered to domesticated parents), and for population (Table S4). To account for nonindependence between observations, we also included female, male, pair, clutch and breeding season identity as random effects, as applicable (Table S5).
We meta-summarized the effects of the inversion genotype (i.e. AD and DD) in two weighted 'lm' models using the R package 'stats'; one model for the reproductive performance traits (1-8) and one for the survival traits (9-13) listed above. As response variables, we used the estimated genotype effects from the mixed models, weighted by the multiplicative inverse of their standard error to account for the level of uncertainty (Rosenthal & DiMatteo, 2001). We removed the intercept and fitted the genotype as the only fixed effect (3 levels).

| Simulating the evolution of the Tgu11 inversion
A detailed description of the simulation and the estimation of parameters following an ODD (Overview, Design concepts and Details) protocol (Grimm et al., 2010) can be found in the Supplementary Methods.
In brief, we used individual-based models to simulate the allele frequency changes over time, given the estimated effects of the inversion haplotypes on fitness-related traits (using the untransformed estimates shown in Table S6). We assumed that the inversion haplotypes followed a Mendelian inheritance without any fitness-related de novo mutations. Each simulation started from an initial population of 400 birds with a D allele frequency of 50% in Hardy-Weinberg equilibrium (i.e. 100 birds of type AA, 200 of type AD and 100 of type DD, with sex randomly assigned). We simulated random mating and discrete nonoverlapping generations, with generation as the time step. Each generation went through phases of embryo production and development from embryo to recruitment. The resulting recruits became the pool of adults for the next generation. For each generation, the frequency of the derived D allele was calculated from all surviving recruits.
To cover the complete reproductive cycle per generation in aviary set-ups ( Figure S2), we not only considered the 13 focal traits described above, but additionally included all possible sources of genotype effects on reproductive performance (i.e. female fertility in aviary, maternal and paternal effects on hatchling survival) and offspring survival (i.e. survival from postindependence to recruitment; Figure S1B) as below.
During the production phase, the number of eggs laid by a female or sired by a male ('gametes') was drawn for each genotype (AA, AD, or DD) from zero-inflated Poisson distributions, based on the empirical proportion of zeros and the mean number of eggs among our captive zebra finches. For each laid egg, we simulated its chance of being fertilized based on the fertility rate of its maternal genotype. Then, we randomly paired the previously simulated genotypes of 'gametes' that were produced by males and females to form diploid embryos. Because we randomly generated gamete numbers for males and females separately, we needed to discard excess embryos that had only one parent assigned, which was done randomly with regard to genotype. The genotype of each embryo was simulated by randomly drawing one allele from each genetic parent.
During the development phase, the survival of each embryo was determined by drawing from a binomial distribution, based on the survival probabilities estimated for the combination of genotypes of the embryo and its parents (Table S6, Figure S2). The survival probability of an individual was calculated as the product of the absolute survival rates (for three survival phases from embryo to recruitment) based on the individual's own genotype (G1a-G2a and G3) and the estimated relative effects on survival based on its mother's (estimates of G1b-G2b) and father's genotype (estimate of G2b); for details, see Supplementary methods. The number of eggs produced by an adult and the probability of survival given a certain genotype were estimated based on the data from the captive zebra finches (Table S6).
First, we ran the simulation described above using empirically es-  Figure S2). Second, we ran simplified simulations of the antagonistic scenario (i.e. Scenarios 3 and 4) to study how the effect size of the recessive deleterious effect influenced the maintenance of the D allele at generation 1000. Thus, we used a fixed additive positive effect on reproductive performance (with selection favouring the D allele, with selection coefficient s Reproductive performance = −0.2 or 1 − ω Reproductive performance DD , such that the relative fitness of the three inversion types for both female fecundity and male siring success is: ω Reproductive performance AA = 1, ω Reproductive performance AD = 1.1, and ω Reproductive performance DD = 1.2), and a variable recessive deleterious effect on survival (selection disfavouring D with s Survival ranging from 0 to 1, such that the relative fitness of the three inversion types for embryo survival to recruitment is: ω Survival AA = 1, ω Survival AD = 1, and ω Survival DD ranging from 0 to 1).
All data and scripts are available through the Open Science Framework.

| Population frequency of the inverted allele
In our captive populations, the allele frequency of the derived allele D ranged from 0.34 to 0.55 (Table S1)

| Additive beneficial effects of the inverted allele on reproductive performance
In comparison with the ancestral genotype AA, the inverted type D significantly increased male and female reproductive performance in an approximately additive manner (Figure 1a

| Recessive deleterious effects of the inverted allele on survival
Inversion type D reduced the probability of survival during early ontogeny in a recessive manner (Figure 1b Table S3 for each estimated effect size). Embryos were less likely to survive if the embryo itself or the genetic mother of the embryo were homozygous for the D allele. Similarly, nestlings with a DD genotype were on average lighter at 8 days of age, less likely to survive to independence, and overall had a shorter lifespan (Figure 1b). Note, however, that none of these detrimental effects reached statistical significance on their own.

| Exploring the range of polymorphism stability
In simplified scenarios of antagonistic pleiotropy, we simulated a fixed additive beneficial effect on reproductive performance (shown in Figure 3b

| DISCUSS ION
Our analyses suggest that the derived inversion on the zebra finch chromosome Tgu11 should initially have been able to spread as a result of its beneficial additive effects on reproductive success (Figure 1a).
Later on, the increase in frequency of the derived allele may have slowed down (Figure 2) by its recessive negative effects on survival in F I G U R E 1 Estimated standardized effect sizes of the Tgu11 inversion genotypes (AD and DD) relative to the homozygous ancestral genotype (AA) representing the baseline phenotype (indicated in white). Effects on male and female reproductive performance traits (a) and on survival traits (b), with their meta-summarized effects. Red indicates positive effects on fitness-related traits, whereas blue indicates negative effects. For each genotype, '+' indicates p < .1, '*' indicates p < .05, '***' indicates p < .0001; no symbol indicates p ≥ .1. All estimates can be found in Tables S3 and S4. DD homozygotes (Figure 1b). However, note that our estimated recessive deleterious effects were not statistically significant on their own.
Additionally, our simulations show that the estimated negative effect on survival was not large enough (Figures 2c, S3C) and would have to be about twice as large (Figures 2d, 3, S3D) to yield equilibrium frequencies that match the observed frequency of 53% in the wild.
We can now speculate about whether the D allele is currently on its way to fixation, or whether we have underestimated the survival cost or overestimated the beneficial effects of the D allele in our captive setting. The first possibility-an ongoing rapid sweep of the D allele-is rather improbable, because the speed of predicted change is relatively high (Figures 2c, S3C, from 50% to 75% in only 44 generations), and no change was observed in captivity across 14 years. Hence, it is more likely that we observe a state that is close to equilibrium. It is also more likely that the magnitude of the fitness effects in the wild differ from those that we estimated in captivity.

F I G U R E 2
Changes in the frequency of the inverted D allele in 50 simulations of a population with 400 adult birds over 1000 generations using estimated total fitness measures (Table S6; combination of female fecundity corrected for fertility in aviaries, male siring success and embryo survival until recruitment corrected for effects from the genotypes of the two parents; Figure S2). We also simulated genetic drift by randomly drawing 400 recruits from the pool of all produced recruits of the previous generation (477 to 841 recruits; see methods for details). Shown are five simulation scenarios illustrating the consequences of ( Irrespective of whether results from captivity can be extrapolated to the wild, our study on the zebra finch chromosome 11 inversion allows us to make an important general point. Weak net heterosis can result from the combination of a weak additive beneficial effect with a recessive deleterious effect (see also Rose, 1985; Zajitschek . One may argue that the probability that these two components come together is low, because beneficial de novo mutations are expected to be rare. However, an inversion can also create a beneficial effect simply by physically linking two alleles that together have a positive fitness effect (two loci in the inversion creating epistasis) and by preventing recombination that would break up the favourable combination Kirkpatrick & Barton, 2006;Villoutreix et al., 2020;Yeaman, 2013).
Thus, whenever a large inversion arises that creates such a beneficial effect, it has a high probability of increasing in frequency in the population (Charlesworth, 2020;Hermisson & Pennings, 2005).
However, given the numerous genes that are captured inside the inversion, the probability that a recessive deleterious mutation is included is also high. Especially in large populations where inbreeding is rare-as in the zebra finch (Balakrishnan & Edwards, 2009;Knief et al., 2015)-selection against recessive deleterious mutations is so inefficient (Charlesworth & Charlesworth, 1987;Charlesworth & Willis, 2009) that numerous recessive deleterious mutations can be found in any genome (Bataillon & Kirkpatrick, 2000). Large inversions (as opposed to small ones) are particularly likely to capture recessive deleterious mutations during the initial establishment of the inversion (Connallon & Olito, 2022;Santos, 1986), but this does not always imply lethality of the inversion homozygotes (DD), as observed in a few iconic examples of inversion polymorphisms (e.g. Küpper et al., 2015;Tuttle et al., 2016). Instead, negative effects may be relatively subtle (i.e. a slightly reduced fitness of DD compared with AA; also see Mérot, Llaurens, et al., 2020, this study). For the polymorphism to be maintained, the recessive deleterious effect can range from mild (leading to a high equilibrium allele frequency of D) to lethal (low D allele frequency; Figure 3), as long as it exceeds the beneficial additive effect. This stability over a broad range of conditions makes such a scenario an appealing explanation for the occurrence of polymorphisms. Note that the deleterious D allele is not easily lost from a population of Ne = 400 individuals, even in the extreme case of complete lethality of the homozygotes (Figure 2, see also Kimura & Ohta, 1969). Only in smaller populations, with more genetic drift, loss of the D allele should happen more frequently (Bataillon & Kirkpatrick, 2000;Kimura & Ohta, 1969). At the other end of the spectrum of effect sizes, weak beneficial effects (c) In the simulations, the D allele had a recessive deleterious effect on survival with a selection coefficient ranging from 0 to 1 (i.e. relative fitness ranges from ω DD = 1 to ω DD = 0). In (a), red highlights the scenarios in which the 95% quantiles of the simulated frequencies of the D allele at generation 1000 overlapped with 50% (dashed line). The blue line in (a) shows the D allele frequency at equilibrium as a function of the selection coefficient against DD concerning survival given the fixed additive beneficial effect on reproductive performance. For selection coefficients against DD that are larger than 1/12, the relative fitness of DD drops below the relative fitness of the AD heterozygote, meaning that both homozygotes are selected against (s AA >0 and s DD >0). In this region of net heterosis, the equilibrium frequency of the D allele equals s AA /(s AA + s DD ) (Falconer & Mackay, 1996). With s AA = 1−(1/1.1) and s DD = 1−((1−s Survival DD )/1.1), we obtain a D allele frequency = 1/(12*s Survival DD ).   Berdan et al., 2021). In contrast, the D allele of chromosome 11 continues to recombine with itself at a rate that is still about half of the genome-wide recombination rate, which should be sufficient to prevent long-term degeneration (Charlesworth et al., 1993;Felsenstein, 1974;Muller, 1964).
Our results highlight that small antagonistic pleiotropic effects on different fitness-related traits can contribute to the maintenance of large chromosomal inversion polymorphisms, derived from a single mutational event. The tight physical linkage of a large number of allelic variants at multiple loci makes it virtually impossible to identify the specific variants causing the fitness effects, but allows to study the genetic effects of the 'inversion allele' on fitness. Inversions are therefore often referred to as 'supergenes' for highly polygenic traits. The sum of several tiny effects across the entire inversion haplotype might still be small and hence only be detectable with large sample sizes.
We observed (a) a small additive beneficial effect on reproductive success (s = 0.1), which supposedly is a relatively rare phenomenon that may have allowed the initial spread of the derived allele and (b) a small recessive deleterious effect on survival (s = 0.05), which is supposedly common (due to the initial capture of a deleterious load by the inversion; Charlesworth & Willis, 2009). Our findings suggest that large chromosomal inversions may link beneficial and deleterious mutations that synergistically prevent the inversion from going extinct or from spreading to fixation. Our study calls for future work that estimates the effect of inversions on multiple fitness-related traits in other species, preferably in the wild.

AUTH O R CO NTR I B UTI O N S
Y.P., W.F. and B.K. conceived the study. Y.P. analysed the data with inputs from W.F. and U.K. Y.P. and W.F. interpreted the results with inputs from U.K. and B.K. Y.P. and W.F. wrote the manuscript with inputs from U.K. and B.K.

ACK N O WLE D G E M ENTS
We

CO N FLI C T O F I NTE R E S T S TATE M E NT
We have no competing interests.

DATA AVA I L A B I L I T Y S TAT E M E N T
Reproductive performance and offspring survival data can be found