A little bit of sex prevents mutation accumulation even in apomictic polyploid plants
In the absence of sex and recombination, genomes are expected to accumulate deleterious mutations via an irreversible process known as Muller’s ratchet, especially in the case of polyploidy. In contrast, no genome-wide mutation accumulation was detected in a transcriptome of facultative apomictic, hexaploid plants of the Ranunculus auricomus complex. We hypothesize that mutations cannot accumulate in flowering plants with facultative sexuality because sexual and asexual development concurrently occurs within the same generation. We assume a strong effect of purging selection on reduced gametophytes in the sexual developmental pathway because previously masked recessive deleterious mutations would be exposed to selection.
We test this hypothesis by modeling mutation elimination using apomictic hexaploid plants of the R. auricomus complex. To estimate mean recombination rates, the mean number of recombinants per generation was calculated by genotyping three F1 progeny arrays with six microsatellite markers and character incompatibility analyses. We estimated the strength of purging selection in gametophytes by calculating abortion rates of sexual versus apomictic development at the female gametophyte, seed and offspring stage. Accordingly, we applied three selection coefficients by considering effects of purging selection against mutations on (1) male and female gametophytes in the sexual pathway (additive, s = 1.000), (2) female gametophytes only (s = 0.520), and (3) on adult plants only (sporophytes, s = 0.212). We implemented recombination rates into a mathematical model considering the three different selection coefficients, and a genomic mutation rate calculated from genome size of our plants and plant-specific mutation rates. We revealed a mean of 6.05% recombinants per generation. This recombination rate eliminates mutations after 138, 204 or 246 generations, depending on the respective selection coefficients (s = 1.000, 0.520, and 0.212).
Our results confirm that the empirically observed frequencies of facultative recombination suffice to prevent accumulation of deleterious mutations via Muller’s ratchet even in a polyploid genome. The efficiency of selection is in flowering plants strongly increased by acting on the haplontic (reduced) gametophyte stage.
KeywordsApomixis Haploid selection Mutation accumulation Polyploidy Plants
The ratio of substitution rates at non-synonymous and synonymous sites
The first filial generation of offspring of different parental species
The fixation index
Thousand years ago
Whole transcriptome shotgun sequencing
Simple sequence repeats
The rate of genomic deleterious mutation accumulation and its effects are fundamental to the evolution of species . The evolution of mutation rate is shaped by the presence or absence of recombination, which is an important consequence of meiosis and sex [2, 3]. Recombining genomes eliminate deleterious mutations in significantly higher rates than is possible in non-recombining genomes [4, 5] because recombination can segregate multiple deleterious mutations into single linkage groups that can be eliminated . Obligate asexual lineages, in contrast, are expected to accumulate recessive deleterious mutations in an irreversible, ratchet-like manner over generations as the least loaded genotypes are lost by random drift (Muller’s ratchet; [7, 8]). In theoretical models, even low recombination rates slow down mutation accumulation [7, 8]. However, empirical studies on genomic mutation accumulation are still rare, especially for polyploid genomes. Polyploidy multiplies the mutation load by cU whereby c is the ploidy level and U is the mutation rate per haploid genome . Polyploid genomes, however, are typical for many asexual eukaryotes and are particularly common in asexually reproducing plants . A novel hypothesis by  for flowering plants predicted that even in high polyploids, low rates of facultative sexuality would suffice to counteract mutation accumulation.
We hypothesize that two factors effectively counteract mutation accumulation in polyploid asexual plants: (1) facultative sexuality and (2) selection on reduced, recombinant sexual gametophytes. Strikingly, a transcriptome study of facultative asexual hexaploid lineages of the Ranunculus auricomus complex revealed that dN/dS ratios did not significantly differ between sexual and asexual species. Asexuals showed no signs of genome-wide mutation accumulation despite ca. 80 kya of hexaploid genome evolution . These hexaploid members of the Ranunculus auricomus species complex exhibit facultative gametophytic aposporous apomixis, where asexual development with unreduced gametophytes runs initially in parallel with the sexual development with reduced gametophytes  (shown in Fig. 1). Evidence of low to medium recombination rates was so far inferred only from natural populations . Here we aimed to estimate (1) the recombination rate per generation using progeny arrays, (2) differential effects of purging selection on reduced/unreduced gametophytes by calculating abortion rates, and (3) the speed of mutation elimination in the hexaploid genome under the observed recombination rate and three selection scenarios via a mathematical model.
Genotyping using microsatellite analysis of three progenies of R. carpaticola × R. cassubicifolius. C = maternal clone, M = SSR mutant clone, R = recombinant
Progeny array (mother plant ID)
No. of progenies
No. of offspring genotypes
No. of C, M and R plants per progeny array
% of recombinant genotypes per progeny arrays
31 (C), 4 (M), 4 (R)
34 (C), 1 (M), 3 (R)
25 (C), 5 (M), 0 (R)
30 (C), 3 (M), 2 (R)
Starting at time t = 0 generations after a new deleterious mutation has become fixed in a non-recombining clone, the frequency of mutant clones κm = 1. A facultative asexual plant population with a non-zero recombination rate per generation and with one fixed deleterious mutation consists of three different offspring classes in the first generation (Additional file 3a). The frequencies of genotypes are given as: ρ0 (non-mutant recombinant from sexual seeds; ‘R’ genotype in Additional file 3a-b), ρm (mutant recombinant from sexual seeds; ‘R’ genotype marked with an asterisk in Additional file 3a-b), κm (mutant clone from apomictic seeds; ‘C’ genotype marked with an asterisk in Additional file 3a-b). We assume that recombination operates in all subsequent generations with the empirically estimated rate r = 0.061 or ~ 6% of recombinant genotypes per generation and that further generations can continue to produce both recombinant and clonal offspring (‘R’ and ‘C’ genotypes in Additional file 3a-b). Hence, in the second generation, the mutant clonal and mutant recombinant mothers each can produce three offspring classes (i.e., non-mutant and mutant recombinants and mutant clones). In contrast, the non-mutant recombinant mother can produce non-mutant clonal offspring (‘C’ in Additional file 3a-b) with a frequency given as κ0 in the second and following generations (Additional file 3c).
If selection s = 1.000, any mutated gametophyte is eliminated. Considering a selection s = 0.520 on female sexual gametophytes only (as approximated by the abortion rate), not only non-mutant recombinants (ρ0 = r|s–2|− 1) but also mutant recombinants (ρm = ρ0|1–s|) might produce offspring. Over generations, the frequency of mutant recombinants decreases with ρm(t) = ρmexp(−ρ0t). Similarly, the frequency of mutant clones κm(t) = (1-ρm)exp(−ρ0t) declines with time:
Any generation with non-zero recombination rate releases non-mutant clones with frequency κ0(t) = 1-[(ρ0 + κm(t))], or:
The speed of Muller’s ratchet under obligate asexuality, assuming fixed population size N = 103 individuals and deleterious mutation rate U = 1.116. n0 = Number of individuals within the least-loaded class, s = selection coefficient, t(J) = interclick time estimate based on Eq. 34 in , t(NS) = interclick time estimate based on Eq. 33 in  and t(ME) = interclick time estimate based on Eq. 24 ; all time estimates are given as numbers of generations per “click”. Bold text = estimates of the selection coefficients of the fastest ratchets; the (J)-value was computed according to Eq. 34 in , corresponding to a U-shaped function with a local minimum at s = 0.212, which we considered as a critical value of the fastest ratchet acting on sporophytes
We present here for the first time a model of mutation elimination in facultative asexual plants under consideration of empirical estimates of recombination rate, selection on gametophytes, and polyploid genomes in hexaploid Ranunculus auricomus. Our model provides support for the hypothesis that low rates of facultative sex prevent genome-wide mutation accumulation even in polyploids, as it was actually observed in a transcriptome study of the same model system .
Recombination rate mostly depends on the degree of facultative sexuality, a factor which considerably varies among flowering plants. In general, plants with alternative meiotic and apomictic development in the same ovule, as it is the case in apospory (Fig. 3) or adventitious embryony, have higher degrees of facultative sexuality (i.e., higher proportions of sexually formed seed in one seed generation). These developmental pathways are found in almost 90% of all apomictic plant genera . Hence, our study can serve as a model system for the great majority of apomictic plants. By using averaged data from different environmental conditions , we covered a broad range of naturally occurring variation in proportions of sexual versus apomictic seed formation. Indeed, natural populations of R. carpaticola × R. cassubicifolius exhibit a comparable genetic variation . Some other apomictic plant genera show developmental pathways where meiosis is either restitutional or mitosis-like (e.g., diplospory ) or with suppressed recombination (e.g., permanent translocation heterozygosity; ). In these cases, alternative sexual pathways cannot be easily realized, because meiosis itself is altered, and there is no developmental phase where sexual and asexual gametophyte development could exist in parallel within one ovule. Hence, these forms of apomixis tend to be more obligate . Indeed, transcriptomes of Oenothera (a rare case of obligate asexuality due to translocation heterozygosity; ) and genomes of diplosporous Boechera (except for conserved coding sites; ) showed signatures of mutation accumulation. These cases are in accordance with our results that maintenance of high levels of facultative sexuality, as in Ranunculus, and certain levels of recombination are crucial for the suppression of mutation accumulation in apomictic plants.
Purging selection during reduced phases of the plant life cycle
Our results support the hypothesis that purifying selection at haplontic stages of the life cycle is an important benefit of sexuality . For plants, the prolonged gametophyte phase with many genes expressed appears to be the main target of purifying selection. This has been shown for pollen  and hypothesized for female gametophytes based on previous developmental studies . Here we demonstrate the same mechanism for female development in R. carpaticola × R. cassubicifolius. We used three selection coefficients to estimate mutation elimination via purging selection. The medium selection coefficient s = 0.520 was taken as an approximation to the fitness decline of ca. 52% during the female sexual pathway, inferred from the observed proportions of functional megaspores to sexually formed seeds (Fig. 3). However, regarding recombination and purging selection acting on male gametophytes as well (e.g., ) and additive effects after fusion of male and female gametes, the efficacy of purifying selection in the sexual pathway can be increased by various mechanisms. Self-pollination is frequent in apomictic plants, probably as a consequence of a breakdown of self-incompatibility systems in polyploids; selection acts for self-fertility especially in pseudogamous apomictic plants . After self-fertilization of the egg cell with mutation carrying pollen, the mutation gets the double dosage in the zygote and consequently in the embryo, and hence will be exposed to selection during the sporophytic phase . This happens not only after self-pollination within the same plant (i.e., within the same flower or between flowers of the same individual) but also after “cross-pollination” between clone-mates of a population, i.e., between genetically identical individuals. Furthermore, gene conversion during meiosis could make the mutation homozygous as well . Gene conversion was considered an important mechanism preventing mutation accumulation in ancient asexual animals . Although we have no data on these mechanisms, we regard a strong additive selection coefficient of s = 1.000 possible.
Mutation rate estimates and Muller’s ratchet
There are multiple factors possibly acting against Muller’s ratchet, e.g. epistatic interactions, beneficial mutations, compensatory mutations and gene conversion. However, the purpose of our model is to illustrate the effect of deleterious mutations. We show that even in the absence of all the above mentioned corrective factors, a bit of recombination suffices to prevent facultative asexuals from mutational meltdown. Concerning the beneficial mutation rates, there are two lines of evidence, the theoretical and experimental. Theoretical models suggest that epistatic effects can effectively halt Muller’s ratchet  or that already a low amount of beneficial mutations might have the same effect . However, in the fast ratchet regime, only a considerable amount of beneficial mutations might prevent an asexual population from extinction . The authors confirmed that a combined effect of beneficial mutations and facultative recombination has not been explored even at the theoretical level, and synthesis of a common model including deleterious and beneficial mutations is missing. A full analysis of this complex situation is clearly beyond the scope of our study. Transcriptome data from R. auricomus revealed only a small amount of outlier genes under positive selection , and hence we assume effects of potentially beneficial mutations as negligible. Another line of evidence, the experiments on model organisms, shows considerable discrepancies not only among studied organisms (yeasts, viruses, Arabidopsis, Daphnia) but also among experimental approaches showing differences between laboratory and field conditions . Some studies suggest negligible rates of beneficial mutations [45, 46], others postulate that a considerable proportion of genomic mutations is beneficial [47, 48]. Observations of extremely high beneficial mutations rates are considered to be experimental artifacts . The possible explanation for the surprisingly high beneficial mutation rates observed by some authors  might reflect specific environmental conditions or experimental setup or specific fitness of founder genotypes . An unexpectedly high ratio of beneficial: deleterious mutations might also result from a low detectability of phenotypic variation due to slightly deleterious mutations .
Our model suggests that facultative sexuality in polyploid Ranunculus auricomus halts Muller’s ratchet, because (1) the least-loaded class is restored in every generation cycle, and (2) mutant frequency decreases with persistent low recombination rates and strength of selection on gametophytes. However, in the obligatory absence of sex, Muller’s ratchet might operate quickly, particularly in very small populations [50, 51]. Considering a small effective population size of hexaploid R. carpaticola × R. cassubicifolius and a relatively high deleterious mutation rate adapted from , we estimated the critical selection coefficient, which is responsible for the fastest ratchet ‘clicks’. This critical value of s approaches 0.212, 0.218 or 0.340, depending on whether computed according to [28, 29], or , respectively. Under these selection scenarios without haplontic phases, and without any recombination, the ratchet could click in the worst case scenario within 32 generations (Table 2).
However, this scenario is unrealistic as facultative sexuality is in plants a persistent mechanism. The genetic and epigenetic control mechanisms of apomixis mostly represent just ectopic or asynchronous mis-expressions of the same genes regulating the sexual pathway . The (epi) genetic control factors of apomixis remain in the heterozygous or hemizygous state with the wild-type (epi) alleles [53, 54]. The apomixis-controlling loci represent large, non-recombinant genomic regions and are inherited as dominant factors . Therefore, apomixis-controlling alleles or epialleles cannot become homozygous. Since expression of apomixis is dependent on allele dosage , apomixis can never become obligate. Apomixis is just superimposed on the sexual pathway . The option of an alternative sexual reproductive pathway, or even the return to obligate sexuality, is present. Interestingly, plant species with a lower degree of asexuality and signatures of mutation accumulation are diploids . The lower mutation rate of diploid genomes might help these lineages to survive with “almost no sex” for some time. For polyploid apomicts – the typical condition for most apomictic plants - maintenance of low levels of facultative sexuality prevents mutation accumulation. Hence, long-term evolution of facultative apomictic plant lineages is probably not limited by the threat of Muller’s ratchet despite polyploidy.
We tested here for mutation accumulation in a typical apomictic plant model system with a big polyploid hybrid genome, but still exhibiting facultative sexuality in both female and male development. We used here the novel approach of combining empirical data and mathematical modelling. Our results confirmed that even in a hexaploid genome, the empirically observed low frequencies of facultative sexuality (c. 6% of recombinants per generation) suffice to prevent accumulation of deleterious mutations via Muller’s ratchet. The efficiency of purging selection is in the sexual pathway of flowering plants strongly increased by acting on the haplontic (meiotically reduced) gametophyte stage, in which many genes are expressed. Hence, previously masked recessive mutations are exposed to purging selection in the gametophyte. Results provide a general explanation for maintenance of a little bit of sex in polyploid asexual flowering plants.
Our model is based on empirical estimates of recombination rate, selection coefficients, and genomic mutation rate for the hexaploid ancient hybrid Ranunculus carpaticola × Ranunculus cassubicifolius, a member of the Eurasian R. auricomus complex [24, 27, 35]. Voucher specimens were deposited in the herbarium of the University of Goettingen (GOET).
Recombination rate per generation
As a proxy for recombination rates, we determined the mean proportion of recombinant genotypes per offspring generation by using microsatellite markers. This approach is most feasible when phenotypic markers for studying segregation are not available . Cytogenetic meiosis studies for assessment of recombination rates are so far not possible in our model system because of the difficulties to observe female plant meiosis directly, and the high chromosome numbers (2n = 48). To estimate a minimum recombination rate, we produced progeny arrays of three allohexaploid mother plants from geographically separated populations belonging to the ancient hybrid Ranunculus carpaticola × Ranunculus cassubicifolius [24, 27, 35].
Ranunculus auricomus is a long-lived plant with a long generation turnover (2–3 years) and hence we could analyze just one generation. We selfed the three mother plants under controlled garden conditions, following the rationale of , and generated 30, 38 and 39 F1-offspring per mother plant (“I”, “T” and “V”, respectively; Table 1). We genotyped 107 plants with six nuclear microsatellite loci previously developed out of the RNA-Seq data of . We used the five SSR loci and protocols from  and one additional locus R2252 (forward primer: TCGGGTTCACCCACTAAATC, annealing temperature = 60 °C; reverse primer CATGGACTAGTTTCCGCCAT, annealing temperature = 60 °C). The plants are hexaploid with up to six alleles per single locus, and allele copy number in a genotype cannot be reliably scored from electropherograms. Hence, we scored the fragments as dominant and recorded presences and absences of each allele across all analyzed individuals in a binary matrix (Additional file 1). In the next step, we compared the F1-offspring multilocus genotypes with their respective mother plant genotypes and estimated the total number of genotypes and genotype diversity D (Nei’s corrected genetic diversity) in the software GENOTYPE/GENODIVE . In the same program, we computed a matrix of pairwise genetic distances between all genotypes assuming the infinite allele model (IAM;  and plotted them as histograms (Additional file 2).
Since the recombination rate has the strongest effect on mutation elimination, overestimates of recombinants per generation would invalidate the model. In SSR studies, this could happen if non-maternal SSR variants that have just arisen from replication slippage  would be scored as recombinants. In fact, hexaploid plants can show a high mutational dynamics of SSR variants . To discriminate true recombinants from non-maternal SSR variants we used three character incompatibility methods. We first used character incompatibility analysis as implemented in the module Jactax of the program PICA 4.0 . Any two characters (alleles of the binary matrix) are incompatible with a hierarchical, tree-like divergence pattern if they show all four character state possibilities in at least four different genotypes A, B, C, D (e.g., A: 1/0, B: 1/1, C: 0/1, D: 0/0) which is a strong signal for recombination . Single novel character states or combinations that are compatible with a hierarchical pattern (e.g., A: 1/0, B: 1/0, C: 0/1, D: 0/1) are a signal for mutations in the marker system. The requirement of at least four different genotypes and two different loci was met in all three progeny arrays (Additional file 2). Character compatibility analysis calculates an initial matrix incompatibility (MI), and subsequent stepwise removal of recombinant genotypes contributing to MI finally identify mutant genotypes with MI = 0 (Additional file 2). Second, the same F1-genotypes were concordantly depicted by a recombination network analysis and visualized within a recombination cycle using the software SplitsTree 4.0 . Here, distance-based incompatible splits are visualized as rectangles, in which recombinants are placed at nodes while mutant genotypes appear as terminal branch-offs (Additional file 2: Figure S2). This network topology was further confirmed with a NeighborNet analysis  for which a statistical test (1000 bootstrap replicates) was conducted (Additional file 2). Finally, we considered genotypes as recombinants only if they were determined by all three character incompatibility methods. We averaged the observed proportions of recombinant F1-offspring within each of the three progeny arrays (i.e., recombination rate per generation, r).
Selection on gametophytes during development is an essential part of our plant-specific model . Previous histological observation of gametophyte development on different apomictic plant species showed various forms and stages of abortive phenotypes and indicated different proportions of sexual vs. apomictic gametophytes during developmental stages [25, 39, 63]. Sexual and apomictic gametophytes develop concurrently in adult plants, they do not differ morphologically, but they have allele dosage-dependent gene expression levels . Hence, we assume that abortion rates of reduced sexual vs. apomictic unreduced gametophytes are a result of the different strength of selection on expressed versus masked deleterious mutations. We re-analyzed published developmental data for the Ranunculus carpaticola × Ranunculus cassubicifolius progenies [35, 65] to estimate the abortion rates during the female gametophyte phase as an indirect indicator of the strength of selection. First, for the stage directly after meiosis, we averaged the proportions of unreduced aposporous initial cells (AIC; see Fig. 1) and reduced functional meiotic megaspores from microscopic observations of ovules (data from ). These data averaged the observed variation under both stressed and unstressed conditions, as they may occur in nature. Second, apomictic versus sexual pathways of seed formation were determined using flow cytometric seed screening, and the averaged proportion of well-developed sexual versus asexual seeds was calculated from mature fruits (data from ). Third, averaged seed germination rates were derived from  and, finally, we assessed the proportion of viable recombinant and clonal offspring based on our estimate of the average recombination rate (see Table 1). Proportions of sexual vs. apomictic development were calculated from proportions of surviving initial cells/seeds/progenies at each developmental step (as only these can continue development) to reveal the actual fitness differences of sexual and apomictic development during the gametophytic and the sporophytic phases (see Fig. 1).
We compared proportions of abortions between different developmental phases, i.e., megaspores, embryos and offspring/seedlings (Fig. 3). As expected, we observed higher abortion rates in the reduced (sexual) than in the unreduced (aposporous apomictic) female gametophyte (Fig. 3). The observed mean abortion rate of ca. 52% in reduced/sexual female gametophytes (Fig. 3), was taken as the argument for a medium selection coefficient (s = 0.520).
However, purifying selection is possible in male, reduced gametophytes as well . Not only female gametophytes are partly aborted in apomictic R. auricomus but also pollen [25, 28, 52], where selection against deleterious mutations acts on the male gametophyte as well. Hence, we also take a proxy of s = c. 0.5 for male development. Since sexual development involves the fusion of male and female gametes that have been both purged from mutations, effects of selection will be additive. This justifies also for testing a strong selection coefficient (s = 1.000). For an estimate of selection on adult plants (i.e., sporophytes only), we computed a theoretical model based on mutation fixation (with estimated finite population size) following , which is outlined below.
Modeling of mutation elimination for polyploids
Deleterious mutation rates in plants are U ≈ 0.1–2.0 per genome and generation (for mutations with mean selection coefficients in the range s ≈ 0.1–0.2) . To estimate deleterious mutation rate in hexaploid Ranunculus auricomus, we multiplied the estimate for diploid Arabidopsis U ≈ 0.1 [1, 52, 66] with the correction factor of 11.16 by which the hexaploid Ranunculus genome size is bigger (G6x = 15.69 pg; 2C) . By comparison, the genome of another small-statured plant Amsinckia spectabilis is four times as large as an average Arabidopsis genome  and it exhibits approximately four times higher U than Arabidopsis . The correction factor of 11.16 we use for Ranunculus already regards a diploid versus a polyploid chromosome complement. We prefer the correction factor based on absolute genome size rather than on ploidy level, because Ranunculus auricomus shows ‘genome downsizing’ in polyploids , i.e., the increase of hexaploid genome size is lower than expected by multiplication of haploid genome size with ploidy level. Genome downsizing is a frequent phenomenon in polyploid flowering plants . A more detailed estimate of polyploid genome evolution is not yet feasible as no completely sequenced reference genome is available for Ranunculus. We did not regard in our model rare beneficial mutations which can also halt Muller’s ratchet  because the Ranunculus transcriptome dataset  included only a small amount of outliers of non-synonymous mutations. Since even these mutations are not necessarily beneficial , we regard the effect of beneficial mutations as negligible. We further did not consider negative epistatic interactions of mutations, which may theoretically increase effects of deleterious mutations, but are relatively uncommon .
We assume a mutation (A) with dominance at 1/3 allele dose (i.e., expressed as Aaa in sexual triploid gametophytes but not as Aaaaaa in hexaploid asexual gametophytes or in sporophytes) due to dosage effects. We assume for sexual development predominant bivalent formation at meiosis, regular segregation, and disomic inheritance, as is typical for allopolyploids in which doubled homologs can pair  because Ranunculus carpaticola × Ranunculus cassubicifolius is an allopolyploid hybrid [24, 73]. We further assume non-overlapping generations and that mutation-carrying and mutation-free offspring are released in constant proportions every generation, resulting in an exponential decay of mutant frequency with time (Additional file 3). We assume a low but constant average recombination rate, exposing mutation-carrying gametes to purging selection. Based on these parameters, we investigated an influence of three selection coefficients.
We estimated the time needed for a slightly deleterious mutation to become fixed in a finite, obligate asexual population from the approximation of Muller’s ratchet inferred by  and compared to other approaches [29, 30]. We consider a small effective population size N = 103 estimated from population genetic studies on natural 6x R. carpaticola × R. cassubicifolius which revealed a high Fst value (0.82) , and from field observations of population size by the authors. The diploid mutation rate U2x = 0.100  was adapted for hexaploid Ranunculus (i.e., U6x = 1.116) using the correction factor of 11.16, based on the ratio between an averaged DNA content of non-replicated holoploid genomes of small-statured model plants (e.g., A. thaliana) and hexaploid R. auricomus genotypes .
We calculated a theoretical selection coefficient with presumably highest effect on the “mutational meltdown” of the population [13, 50, 74], inferred from the critical range of selection coefficients which favor the fastest ‘clicking’ of Muller’s ratchet. The critical selection coefficient was computed with the parameter set above as s = 0.212. We consider it as an approximation of selection against deleterious mutations in hexaploid sporophytes only. We regarded the parameter regime of λ = U/s (=1.116/0.212) ≈ 5.3 as the ‘worst scenario’, where Muller’s ratchet clicks very frequently in a small obligatory asexual population (N = 103 plant individuals). We inferred the ratchet speed based on several published approximations (Table 2).
We thank two referees for valuable comments; Jennifer Krüger, Natalia Woźniak, Silvia Friedrichs for technical help, the German Research Foundation for funding (DFG project Ho 4395/4-1 to E.H.). We acknowledge support by the Open Access Publication Funds of the Goettingen University.
EH, LH, TS: study design. LH, EH, SK, DH: data collection and analysis. All authors: writing. All authors have read and approved the manuscript.
This work was funded by German Research Fund “Deutsche Forschungsgemeinschaft” (DFG project Ho 4395/4–1) to E.H. The funders had no role in study design, data collection and analysis, and preparation of the manuscript.
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The authors declare that they have no competing interests.
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