BACKGROUND
finding explanations for the evolution and maintenance of sexual reproduction is a long-running research problem in evolutionary biology.
a large number of scenarios and models have been developed under which either sexual or asexual reproduction could have advantages or disadvantages  <cit> .
in the context of this ongoing search, unisexual  vertebrates are of great interest, because they put popular explanations like muller's ratchet  <cit> , mutational deterministic hypothesis  <cit>  and the red queen  <cit>  to a test.
all unisexual vertebrates are of interspecific hybrid origin  <cit> .
they therefore have similar genome sizes , are exposed to similar environments as their sexual parental species and often directly compete against them.
this allows for the comparison of the two reproductive modes under similar ecological conditions.
whereas the theories of mutation accumulation and selection against deleterious mutations are well established for asexual and sexual species, they are not for unisexual vertebrates that reproduce by hybridogenesis, a reproductive mode that is intermediate between sexual and asexual reproduction.
figure  <dig> illustrates this intermediate position for the typical vertebrate case.
parthenogenesis  and gynogenesis  both represent asexual reproduction, where offspring arise clonally from diploid eggs of an all-female species  that in the past originated from hybridization between two sexual species .
in parthenogenesis the eggs develop by themselves, whereas in gynogenesis their development must be triggered by sperm from a male of one of the hybrid's ancestral parental species .
hence, gynogenetic females must mate, but the male's genome  does not show up in the offspring.
hybridogenesis  resembles gynogenesis in that females of hybrid origin  need males for successful reproduction.
in contrast to gynogens, however, hybridogenetic females discard the ancestral paternal genome prior to meiosis , produce haploid eggs with an unrecombined maternal genome  and then restore diploidy  in their offspring by backcrossing with males of the parental species whose genome was discarded.
in this inclusion of the paternal genome they resemble true bi-sexual species .
thus, the genome of a hybridogen is "hemiclonal", consisting of a clonally inherited maternal part and a sexually inherited paternal part, with no recombination between them  <cit> .
the long time evolutionary perspectives of such hybridogenetic unisexuals have been questioned by several authors  based on the argument that clonal inheritance of a part of their genome exposes them to the same perils of reduced genetic diversity as parthenogenetic or gynogenetic species.
however, hybridogenetic reproduction has not been modelled yet in terms of deleterious mutation accumulation dynamics and susceptibility to drift effects.
given the low number of unisexual vertebrates  and hybridogenetic vertebrates are all-female species , whereas sexual species consist of males and females.
this becomes important when mutation rates are sex-specific.
starting with haldane  <cit>  a number of studies have shown higher mutation rates in males than in females.
current reviews on male/female mutation rates  list ratios in the range from  <dig> to  <dig> for primates, rodents, birds and humans, but data are still very scarce  <cit> .
the male bias has been interpreted as evidence that new mutations occur during dna replication, as spermatogonia divide throughout the whole life of males whereas oogenesis in females is largely complete at birth.
therefore, it has been suggested that evolution is "male driven".
since direct measures of sex specific mutations rates are difficult to obtain, miyata et al.
<cit>  proposed an indirect method of testing for sex differences in mutation rates, namely comparing the evolutionary rate of sex chromosomes and autosomes.
an x chromosome has spent about 1/ <dig> of its history in males, whereas autosomes spend about an equal time in both sexes, and y chromosomes only occur in males.
if male and female mutation rates are different, we expect different evolutionary rates for x and y chromosomes and autosomes.
redfield  <cit>  modelled the effect of elevated male mutation rates on the mutational load in infinite sexual populations and compared the results with those from infinite diploid asexual populations.
she showed that the cost of male mutations can easily exceed the benefit from recombination if populations are sufficiently large.
since hybridogenetic unisexuals are often in direct competition with their sexual parental species due to the forced coexistence, such systems allow testing the effects of sex-specific mutations rates on the relative success of sexual versus asexual reproduction.
we therefore expanded redfield's model for infinite populations and included hybridogenetic reproduction into the comparison.
furthermore, we developed monte-carlo simulations for finite sexual, diploid asexual and hybridogenetic populations of two different sizes  to account for the effect of higher male mutation rates and stochastic events like drift effects and muller's ratchet.
details of the models used in this paper are described in the methods section.
all the three model populations investigated  share the following common features:
- generations do not overlap
- mating is random
- males and females can mate several times
- individuals accumulate new mutations between birth and reproduction.
the distribution of these mutations follows a poisson distribution with the mean of u
- all mutations have the same character of dominancy and the same effect on fitness, regardless of the locus where they occur.
hence they are assumed to be co-dominant.
- all populations  are diploid.
- a homozygous mutation, i.e.
with  <dig> mutated alleles at the same locus, has the same fitness effect as  <dig> heterozygous loci bearing a mutated allele.
- sexual individuals show mendelian recombination
- hybridogenetic individuals do not recombine
- all hybridogenetic populations  live in sympatry with an infinite sexual population maintained in mutation-selection balance.
- sympatric sexual and hybridogenetic populations show the same sex specific mutation rates and both populations show the same type of mutation interaction.
- new mutations in a hybridogen are equally likely to occur on the sexual or on the clonal part of the hybridogen's genome
all three reproduction modes  were compared under the same three population sizes  and the same three types of mutation interaction with different levels of epistasis: independent, quadratic and truncation selection .
RESULTS
since the results for large populations with  <dig> individuals are almost identical to those for infinite populations , they will be presented together
infinite and large populations
figures 3a1–a <dig> and 3b1–b <dig> show the results of the test runs for the infinite and the large populations.
the results correspond well with those of kimura & maruyama  <cit>  and redfield  <cit>  who modeled the effects of epistasis on mutational load in relation to sexual and asexual reproduction.
if mutations show no synergistic epistasis  and male and female mutation rates are the same , all reproductive modes perform equally.
with increasing male to female mutation rate ratio α, asexual reproduction becomes favorable compared to sexual reproduction.
hybridogenetic populations show an intermediate mutation load between asexual and sexual populations, whereas the clonally transmitted genomes of hybridogens accumulate the same number of deleterious mutations as the genomes of an infinite asexual population .
under independent selection, asexual reproduction was the only reproductive mode where large finite populations would suffer from fixations of mutations.
in four out of ten runs the model population ended with one fixed mutation at a locus and in three cases with two fixed mutations
with both the quadratic and truncation forms of synergistic epistasis , when α is approximately larger than  <dig>  the overall ordering of fitness relationships between sexuals and asexuals remains unaltered.
however, as α approaches a value of  <dig>  we have a reversal, whereby sexual reproduction shows an advantage over hybridogenetic and asexual reproduction.
this advantage becomes larger with stronger synergistic epistasis ; an observation that is consistent with the mutational deterministic hypothesis  <cit> .
with α =  <dig> and high levels of synergistic epistasis, recombination is more effective in reducing the mutation load; consequently, sexual reproduction is more advantageous than asexual reproduction  <cit> .
note that for α =  <dig>  the advantage of sexual reproduction would be also more pronounced if mutation rates u were to be increased  <cit> .
for the epistatic cases with α =  <dig>  hybridogens show behavior that is generally intermediate between the asexuals and the sexuals.
this is because they have an advantage over the asexuals, since the sexual part of their genome – derived form the parental species – has been subject to the purifying effects of recombination.
meanwhile, as α becomes larger, they do better than the sexuals because the clonal half of their genome is not subject to the increased accumulation of mutations incurred in their sexual counterpart.
furthermore, an interesting – and somewhat counterintuitive – aspect of the simulation results is the observation that the mutational load on the clonally transmitted part of the hybrid genome does not increase, but rather is reduced as α increases.
in effect, with truncation selection, hybridogenetic populations show a higher average fitness than asexual populations for all tested α values, and a higher fitness then sexual populations for α ≥  <dig> the more the paternal genome is loaded with mutations relative to the maternal genome, the more truncation selection prevents the accumulation of mutations on the clonally transmitted part of the hybridogenetic genome.
at high α values, the clonal part of the genome is virtually free of deleterious mutations.
under asexual reproduction, the nonexistent selection against the first few mutations per individual led to the fixation of the maximum number of  <dig> allowed mutations in the finite populations, whereas under sexual reproduction no fixations occurred.
with hybridogenetic reproduction, the number of fixed mutations decreases from  <dig> at α =  <dig>  through  <dig> at α =  <dig> to  <dig> at higher α values.
small populations
small populations  are prone to accumulate higher numbers of mutations due to increased drift effects.
this effect can be seen well in the case of independent mutational effects on fitness .
all asexual test populations reached the maximum number of mutations the computer simulation could handle , and the average population fitness plunged to a level which would not allow persistence anymore .
sexual reproduction successfully prevented the fixation of mutations with mean population fitnesses in the range of the larger and infinite populations.
in hybridogenetically reproducing populations, the clonally transmitted part of the hybrid genomes suffered from substantial fixation of mutations in the small populations .
the resulting low clonal fitness leads to a strongly diminished average fitness in hybridogens as well.
under quadratic selection , mutation accumulation and fixation in small asexually reproducing populations is not as serious as under independent selection but still worse than with the other two reproductive modes .
sexual populations seem to be largely resistant to drift effects, since even at a size of  <dig> they do not differ much from infinite populations in their average fitness, and no mutations got fixed within  <dig> generations.
susceptibility of hybridogenetic populations to drift effects varies with male to female mutation rates: averaged over the ten model populations the number of fixed mutations decreased from  <dig>  and  <dig>  at α =  <dig> and α =  <dig>  respectively, through  <dig>   to  <dig> .
despite the occasional fixation of mutations, small hybridogenetic populations showed a higher average fitness at high α values than small sexual populations.
the clonally transmitted hybrid genomes showed reduced fitness at α =  <dig>   <dig> due to the fixed mutations; but at higher male to female mutation rate ratios, the strong selection against new mutations prevented these genomes from accumulating and fixing mutations and, hence, resulted in high fitness.
under truncation selection, the nonexistent selection against low numbers of deleterious mutations led to the accumulation and fixation of the maximum allowed number of  <dig> mutations in all asexual model populations; but nevertheless, the average mean population fitness did not degrade compared to the large and the infinite populations, because the fixed mutations do not cause a decline in fitness.
no fixation of mutations occurred in the sexual populations, and again population size had only little effect on the average population fitness .
although hybridogens show similar levels of fitness as asexuals for all α values, they fix fewer mutations on their clonal genomes than asexuals do for all tested α >  <dig>  namely on average  <dig> ,  <dig>  and  <dig>  for α =  <dig>   <dig> and  <dig>  respectively, compared to  <dig> in asexuals.
).
speed of muller's ratchet
since we recorded the losses of least loaded classes and the fixation of mutations during the model runs, this allowed us to determine the speed of muller's ratchet.
finite populations of the larger size  only occasionally showed fixations of mutations  but in smaller populations, fixations occurred frequently enough to allow a comparison of the speed of muller's ratchet between asexual reproduction and hybridogenetic reproduction with α =  <dig>  in this comparison, both model populations had the same genome size and genomic mutation rate.
figure  <dig> shows the average of  <dig> test runs per reproductive mode and type of mutation interaction.
the fixation of mutations followed the loss of the least loaded classes closely, regardless of the type of reproduction, which confirms the findings of charlesworth & charlesworth  <cit> .
under hybridogenetic reproduction, muller's ratchet not only clicks significantly slower than under asexual reproduction; the level of fixed mutations at which the ratchet would come to a near halt  is also lower.
discussion
longtime evolutionary perspectives for hybridogenetically reproducing organisms often have been considered as not being very promising  but knowledge of the dynamics of deleterious mutation accumulation and selection in hybridogens has been limited so far.
since one half of the hybridogens' genome is passed on without recombination from generation to generation, it is understandable that the mechanisms of mutation accumulation have been expected to operate in a similar manner as in asexual populations .
however, this expectation does not necessarily hold.
as the results in this study indicate, the dynamics of mutation accumulation in hybridogens can differ from the ones in asexual populations; both in terms of the speed of mutation accumulation and   the total number of deleterious mutations.
the fitness effects of deleterious mutation accumulation in hybridogens are not simply at an intermediate level between sexual and asexual populations.
the latter comparison depends on the ratio of male to female mutation rates α. if α =  <dig> in a large population with synergistic epistasis, conventional expectations hold and the average fitness of a hybridogenetic population is indeed intermediate between sexuals and asexuals.
this intermediate position is due to the property that mutation accumulation is slower within the chromosomes derived from the sexually reproducing parental species .
however, the situation starts to change as α becomes larger than a value of one.
in the latter case, sexually reproducing species not only face the often cited twofold costs of reproduction  <cit> , but also suffer from a higher average mutation rate than comparable all female asexual species .
since hybridogenetic organisms do not recombine, the mutations originating on the sexual parent's gamete are not passed on to the next generation.
whereas gametes of sexuals contain, on average, half of the paternally inherited, half of the maternally inherited mutations plus half of the newly acquired mutations, the female gametes of hybridogens contain only the maternally inherited mutations plus the newly acquired mutations that occurred on the clonally transmitted part of the genome.
to understand the simulation results presented here, it is necessary not only to consider the mechanistic differences in modes of genetic transmission, as represented by the three reproductive systems discussed here.
it is also necessary to understand the interplay of population dynamics with these mechanisms.
this becomes apparent when one considers mutation accumulation in the clonal part of the hybrid genome.
as seen in the cases with quadratic and truncation selection , as α becomes larger, mutation accumulation goes towards zero in the clonal part of the hybrid genome .
this is not because mutations do not occur on the clonal part, but rather that they accumulate much faster on the sexual part of the genome, and hence the quadratic and truncation selection effects allow for no more mutations to accumulate on the whole genome.
selection is too effective in purging the "late arriving" mutants on the clonal part of the genome.
in other words, if all mutations that a hybridogen carries affect fitness, regardless whether they reside on the sexually or clonally transmitted part of the genome, selection against all newly acquired mutations is strong if mutations interact synergistically, because of the "borrowed" high mutational load on the paternal genome.
since only the maternal genome can be transmitted , the clonally transmitted part of the genomes of hybridogenetic organisms stays remarkably  free of deleterious mutations.
the higher the male to female mutation rate ratio in the sexual parent, the more pronounced is this "shielding effect" in hybridogens.
in large populations with truncation selection, this effect can be strong enough that even at an α approaching  <dig>  the fitness of the hybridogenetic population can be higher than its asexual counterpart .
a challenge for the persistence of asexually and hemiclonally reproducing organisms are drift effects in small or fragmented populations  <cit> .
the comparison of the speed of muller's ratchet between asexual and hybridogenetic populations  showed an effective reduction in the ratchet speed for hybridogenetic populations.
this advantage originates from the model properties that only half of a hybridogen's new mutations occur on its clonally transmitted part and, thus, can be transmitted to the next generation, whereas in an asexual organism all novel mutations are transmitted to the offspring.
in this model, fixation of mutations followed the loss of the least loaded fitness classes in asexual populations  and hybridogenetic populations .
although population size had little influence on the mutational load and on the number of fixed alleles in the size-limited populations with epistatic mutation interaction, hybridogenetically reproducing populations were generally less prone to the accumulation of deleterious mutations than asexual populations.
the selective forces against deleterious mutations seemed to be sufficiently effective to prevent a substantial drop in average population fitness , as the mutation rate on the clonally transmitted genome is reduced relative to that on the sexual genome.
even in quite small populations, sexually reproducing organisms are at a disadvantage compared to all female asexual populations if the male to female mutation rate ratio is high, at least under quadratic and truncation selection .
this result could be due to our choice of the low rate of  <dig>  deleterious mutations per female and generation in the asexual populations.
however, the effective values of u are still debated , and the value chosen for our simulations seems to be somewhere in the middle of the reported ranges.
furthermore, our u-value of  <dig>  allows a direct comparison with the results of redfield  <cit> .
at high levels of α, small hybridogenetic populations accumulate fewer mutations than small populations of their sexual parent species if mutations interact synergistically.
since hybridogenetic populations are better protected against the accumulation and fixation of deleterious mutations than asexual populations, and at the same time, are less affected by the negative impacts of a higher deleterious mutation rates in males, hybridogenetically reproducing organisms perform better than asexual and sexual organisms at reduced population sizes under quadratic and truncation selection if α >  <dig>  this result is quite interesting as this is the area, where the average u is between  <dig>  and  <dig> , and, therefore, in the range where sexual reproduction becomes favorable over asexual reproduction in mutational deterministic models  <cit> .
CONCLUSIONS
the simulation results in this work indicate that with male to female mutation rates higher than one , hybridogenetic populations can be less prone to the accumulation of deleterious mutations, and hence have a higher population fitness than their sexual counterpart.
as α becomes larger, mutation accumulation in the part of the genome derived form the sexual species is faster, and these mutations make it more difficult for new mutations to accumulate on the clonal part of the genome .
this effect becomes more pronounced with synergistic forms of epistasis .
furthermore, in cases with epistasis and α =  <dig>  although sexual populations have a higher fitness than the hybrids, the hybrids in turn do better than the asexuals, because mutations are purged more effectively in the sexually derived part of their genome.
these results lead to the question why hybridogenetic organisms are so scarce compared to sexual species.
at the genetic level, one reason could be that, in the real world, mutation interactions are far more complicated and diverse than the ones used in this model  <cit> .
if we assume that mutation interactions vary from negative through no to positive epistasis, with an average effect of no epistasis, sexual reproduction still acts best against the accumulation of mutations in very small populations .
also, in natural populations, neither selection nor dominance coefficients are constant values for all deleterious mutations.
another reason why hybridogens are scarce could be that the real obstacle is actually to achieve hybridogenetic reproduction  <cit> .
in order to do so, meiosis has to be circumvented or disrupted which often leads to infertility or sterility  <cit> .
furthermore, one cannot consider the evolution of hybridogenesis by only looking at genetic mechanisms but has to include ecology as well.
the survival of the hybrid population totally depends on the continued contact with the sexual parental population which serves as an indispensable sperm donor.
their sperm-dependence prohibits the hybridogens to occupy niches which are markedly different from the niche of the sexual parent and does not allow out-competing the parental species and leading it to extinction.
hence, hybridogenesis can only evolve and persist under ecological conditions that allow stable populations of both, the sexual parasite  and their sexual hosts .
this ecological issue was not considered in the present work, but has been dealt with in previous publications  <cit>
our result, that clonal or hemiclonal polymorphism can not be maintained with the used model framework, suggests that the often observed diversity in sympatric clones or hemiclones either originates from subsequent recruitments of new clones or hemiclones through repeated primary hybridization, or that diversity in clones is maintained through the occupancy of different microhabitats or "frozen niches"  <cit>
although mutation accumulation is less of a threat to hybridogenetic populations  under several of the conditions used in this model, there are other factors inherent to asexual or hybridogenetic reproduction which affect the long time success of such populations: the response to rapidly changing environments or to parasites is still best ensured through sexual reproduction  <cit> .
but nevertheless, the results of this study indicate that, as suggested by schultz  <cit> , hybridogenetic species could in fact act as a transition state in the formation of new sexual species once recombination is reestablished and reproductive isolation from both sexual ancestors occurred.
the surprisingly small accumulation of deleterious mutations on the clonally transmitted part of a hybridogen's genome under synergistic epistasis and differential sex specific mutation rates supports this as an option for new species formations.

 finding explanations for the evolution and maintenance of sexual reproduction is a long-running research problem in evolutionary biology. a large number of scenarios and models have been developed under which either sexual or asexual reproduction could have advantages or disadvantages. in the context of this ongoing search, unisexual vertebrates are of great interest, because they put popular explanations like muller's ratchet, mutational deterministic hypothesis and the red queen to a test. all unisexual vertebrates are of interspecific hybrid origin. they therefore have similar genome sizes, are exposed to similar environments as their sexual parental species and often directly compete against them. this allows for the comparison of the two reproductive modes under similar ecological conditions. whereas the theories of mutation accumulation and selection against deleterious mutations are well established for asexual and sexual species, they are not for unisexual vertebrates that reproduce by hybridogenesis, a reproductive mode that is intermediate between sexual and asexual reproduction. figure illustrates this intermediate position for the typical vertebrate case. parthenogenesis and gynogenesis both represent asexual reproduction, where offspring arise clonally from diploid eggs of an all-female species that in the past originated from hybridization between two sexual species. in parthenogenesis the eggs develop by themselves, whereas in gynogenesis their development must be triggered by sperm from a male of one of the hybrid's ancestral parental species. hence, gynogenetic females must mate, but the male's genome does not show up in the offspring. hybridogenesis resembles gynogenesis in that females of hybrid origin need males for successful reproduction. in contrast to gynogens, however, hybridogenetic females discard the ancestral paternal genome prior to meiosis, produce haploid eggs with an unrecombined maternal genome and then restore diploidy in their offspring by backcrossing with males of the parental species whose genome was discarded. in this inclusion of the paternal genome they resemble true bi-sexual species. thus, the genome of a hybridogen is "hemiclonal", consisting of a clonally inherited maternal part and a sexually inherited paternal part, with no recombination between them. the long time evolutionary perspectives of such hybridogenetic unisexuals have been questioned by several authors based on the argument that clonal inheritance of a part of their genome exposes them to the same perils of reduced genetic diversity as parthenogenetic or gynogenetic species. however, hybridogenetic reproduction has not been modelled yet in terms of deleterious mutation accumulation dynamics and susceptibility to drift effects. given the low number of unisexual vertebrates and hybridogenetic vertebrates are all-female species, whereas sexual species consist of males and females. this becomes important when mutation rates are sex-specific. starting with haldane a number of studies have shown higher mutation rates in males than in females. current reviews on male/female mutation rates list ratios in the range from to for primates, rodents, birds and humans, but data are still very scarce. the male bias has been interpreted as evidence that new mutations occur during dna replication, as spermatogonia divide throughout the whole life of males whereas oogenesis in females is largely complete at birth. therefore, it has been suggested that evolution is "male driven". since direct measures of sex specific mutations rates are difficult to obtain, miyata et al. proposed an indirect method of testing for sex differences in mutation rates, namely comparing the evolutionary rate of sex chromosomes and autosomes. an x chromosome has spent about 1/ of its history in males, whereas autosomes spend about an equal time in both sexes, and y chromosomes only occur in males. if male and female mutation rates are different, we expect different evolutionary rates for x and y chromosomes and autosomes. redfield modelled the effect of elevated male mutation rates on the mutational load in infinite sexual populations and compared the results with those from infinite diploid asexual populations. she showed that the cost of male mutations can easily exceed the benefit from recombination if populations are sufficiently large. since hybridogenetic unisexuals are often in direct competition with their sexual parental species due to the forced coexistence, such systems allow testing the effects of sex-specific mutations rates on the relative success of sexual versus asexual reproduction. we therefore expanded redfield's model for infinite populations and included hybridogenetic reproduction into the comparison. furthermore, we developed monte-carlo simulations for finite sexual, diploid asexual and hybridogenetic populations of two different sizes to account for the effect of higher male mutation rates and stochastic events like drift effects and muller's ratchet. details of the models used in this paper are described in the methods section. all the three model populations investigated share the following common features: - generations do not overlap - mating is random - males and females can mate several times - individuals accumulate new mutations between birth and reproduction. the distribution of these mutations follows a poisson distribution with the mean of u - all mutations have the same character of dominancy and the same effect on fitness, regardless of the locus where they occur. hence they are assumed to be co-dominant. - all populations are diploid. - a homozygous mutation, i.e. with mutated alleles at the same locus, has the same fitness effect as heterozygous loci bearing a mutated allele. - sexual individuals show mendelian recombination - hybridogenetic individuals do not recombine - all hybridogenetic populations live in sympatry with an infinite sexual population maintained in mutation-selection balance. - sympatric sexual and hybridogenetic populations show the same sex specific mutation rates and both populations show the same type of mutation interaction. - new mutations in a hybridogen are equally likely to occur on the sexual or on the clonal part of the hybridogen's genome all three reproduction modes were compared under the same three population sizes and the same three types of mutation interaction with different levels of epistasis: independent, quadratic and truncation selection.  since the results for large populations with individuals are almost identical to those for infinite populations, they will be presented together infinite and large populations figures 3a1–a and 3b1–b show the results of the test runs for the infinite and the large populations. the results correspond well with those of kimura & maruyama and redfield who modeled the effects of epistasis on mutational load in relation to sexual and asexual reproduction. if mutations show no synergistic epistasis and male and female mutation rates are the same, all reproductive modes perform equally. with increasing male to female mutation rate ratio α, asexual reproduction becomes favorable compared to sexual reproduction. hybridogenetic populations show an intermediate mutation load between asexual and sexual populations, whereas the clonally transmitted genomes of hybridogens accumulate the same number of deleterious mutations as the genomes of an infinite asexual population. under independent selection, asexual reproduction was the only reproductive mode where large finite populations would suffer from fixations of mutations. in four out of ten runs the model population ended with one fixed mutation at a locus and in three cases with two fixed mutations with both the quadratic and truncation forms of synergistic epistasis, when α is approximately larger than the overall ordering of fitness relationships between sexuals and asexuals remains unaltered. however, as α approaches a value of we have a reversal, whereby sexual reproduction shows an advantage over hybridogenetic and asexual reproduction. this advantage becomes larger with stronger synergistic epistasis; an observation that is consistent with the mutational deterministic hypothesis. with α = and high levels of synergistic epistasis, recombination is more effective in reducing the mutation load; consequently, sexual reproduction is more advantageous than asexual reproduction. note that for α = the advantage of sexual reproduction would be also more pronounced if mutation rates u were to be increased. for the epistatic cases with α = hybridogens show behavior that is generally intermediate between the asexuals and the sexuals. this is because they have an advantage over the asexuals, since the sexual part of their genome – derived form the parental species – has been subject to the purifying effects of recombination. meanwhile, as α becomes larger, they do better than the sexuals because the clonal half of their genome is not subject to the increased accumulation of mutations incurred in their sexual counterpart. furthermore, an interesting – and somewhat counterintuitive – aspect of the simulation results is the observation that the mutational load on the clonally transmitted part of the hybrid genome does not increase, but rather is reduced as α increases. in effect, with truncation selection, hybridogenetic populations show a higher average fitness than asexual populations for all tested α values, and a higher fitness then sexual populations for α ≥ the more the paternal genome is loaded with mutations relative to the maternal genome, the more truncation selection prevents the accumulation of mutations on the clonally transmitted part of the hybridogenetic genome. at high α values, the clonal part of the genome is virtually free of deleterious mutations. under asexual reproduction, the nonexistent selection against the first few mutations per individual led to the fixation of the maximum number of allowed mutations in the finite populations, whereas under sexual reproduction no fixations occurred. with hybridogenetic reproduction, the number of fixed mutations decreases from at α = through at α = to at higher α values. small populations small populations are prone to accumulate higher numbers of mutations due to increased drift effects. this effect can be seen well in the case of independent mutational effects on fitness. all asexual test populations reached the maximum number of mutations the computer simulation could handle, and the average population fitness plunged to a level which would not allow persistence anymore. sexual reproduction successfully prevented the fixation of mutations with mean population fitnesses in the range of the larger and infinite populations. in hybridogenetically reproducing populations, the clonally transmitted part of the hybrid genomes suffered from substantial fixation of mutations in the small populations. the resulting low clonal fitness leads to a strongly diminished average fitness in hybridogens as well. under quadratic selection, mutation accumulation and fixation in small asexually reproducing populations is not as serious as under independent selection but still worse than with the other two reproductive modes. sexual populations seem to be largely resistant to drift effects, since even at a size of they do not differ much from infinite populations in their average fitness, and no mutations got fixed within generations. susceptibility of hybridogenetic populations to drift effects varies with male to female mutation rates: averaged over the ten model populations the number of fixed mutations decreased from and at α = and α = respectively, through to. despite the occasional fixation of mutations, small hybridogenetic populations showed a higher average fitness at high α values than small sexual populations. the clonally transmitted hybrid genomes showed reduced fitness at α = due to the fixed mutations; but at higher male to female mutation rate ratios, the strong selection against new mutations prevented these genomes from accumulating and fixing mutations and, hence, resulted in high fitness. under truncation selection, the nonexistent selection against low numbers of deleterious mutations led to the accumulation and fixation of the maximum allowed number of mutations in all asexual model populations; but nevertheless, the average mean population fitness did not degrade compared to the large and the infinite populations, because the fixed mutations do not cause a decline in fitness. no fixation of mutations occurred in the sexual populations, and again population size had only little effect on the average population fitness. although hybridogens show similar levels of fitness as asexuals for all α values, they fix fewer mutations on their clonal genomes than asexuals do for all tested α > namely on average, and for α = and respectively, compared to in asexuals.). speed of muller's ratchet since we recorded the losses of least loaded classes and the fixation of mutations during the model runs, this allowed us to determine the speed of muller's ratchet. finite populations of the larger size only occasionally showed fixations of mutations but in smaller populations, fixations occurred frequently enough to allow a comparison of the speed of muller's ratchet between asexual reproduction and hybridogenetic reproduction with α = in this comparison, both model populations had the same genome size and genomic mutation rate. figure shows the average of test runs per reproductive mode and type of mutation interaction. the fixation of mutations followed the loss of the least loaded classes closely, regardless of the type of reproduction, which confirms the findings of charlesworth & charlesworth. under hybridogenetic reproduction, muller's ratchet not only clicks significantly slower than under asexual reproduction; the level of fixed mutations at which the ratchet would come to a near halt is also lower. discussion longtime evolutionary perspectives for hybridogenetically reproducing organisms often have been considered as not being very promising but knowledge of the dynamics of deleterious mutation accumulation and selection in hybridogens has been limited so far. since one half of the hybridogens' genome is passed on without recombination from generation to generation, it is understandable that the mechanisms of mutation accumulation have been expected to operate in a similar manner as in asexual populations. however, this expectation does not necessarily hold. as the results in this study indicate, the dynamics of mutation accumulation in hybridogens can differ from the ones in asexual populations; both in terms of the speed of mutation accumulation and the total number of deleterious mutations. the fitness effects of deleterious mutation accumulation in hybridogens are not simply at an intermediate level between sexual and asexual populations. the latter comparison depends on the ratio of male to female mutation rates α. if α = in a large population with synergistic epistasis, conventional expectations hold and the average fitness of a hybridogenetic population is indeed intermediate between sexuals and asexuals. this intermediate position is due to the property that mutation accumulation is slower within the chromosomes derived from the sexually reproducing parental species. however, the situation starts to change as α becomes larger than a value of one. in the latter case, sexually reproducing species not only face the often cited twofold costs of reproduction, but also suffer from a higher average mutation rate than comparable all female asexual species. since hybridogenetic organisms do not recombine, the mutations originating on the sexual parent's gamete are not passed on to the next generation. whereas gametes of sexuals contain, on average, half of the paternally inherited, half of the maternally inherited mutations plus half of the newly acquired mutations, the female gametes of hybridogens contain only the maternally inherited mutations plus the newly acquired mutations that occurred on the clonally transmitted part of the genome. to understand the simulation results presented here, it is necessary not only to consider the mechanistic differences in modes of genetic transmission, as represented by the three reproductive systems discussed here. it is also necessary to understand the interplay of population dynamics with these mechanisms. this becomes apparent when one considers mutation accumulation in the clonal part of the hybrid genome. as seen in the cases with quadratic and truncation selection, as α becomes larger, mutation accumulation goes towards zero in the clonal part of the hybrid genome. this is not because mutations do not occur on the clonal part, but rather that they accumulate much faster on the sexual part of the genome, and hence the quadratic and truncation selection effects allow for no more mutations to accumulate on the whole genome. selection is too effective in purging the "late arriving" mutants on the clonal part of the genome. in other words, if all mutations that a hybridogen carries affect fitness, regardless whether they reside on the sexually or clonally transmitted part of the genome, selection against all newly acquired mutations is strong if mutations interact synergistically, because of the "borrowed" high mutational load on the paternal genome. since only the maternal genome can be transmitted, the clonally transmitted part of the genomes of hybridogenetic organisms stays remarkably free of deleterious mutations. the higher the male to female mutation rate ratio in the sexual parent, the more pronounced is this "shielding effect" in hybridogens. in large populations with truncation selection, this effect can be strong enough that even at an α approaching the fitness of the hybridogenetic population can be higher than its asexual counterpart. a challenge for the persistence of asexually and hemiclonally reproducing organisms are drift effects in small or fragmented populations. the comparison of the speed of muller's ratchet between asexual and hybridogenetic populations showed an effective reduction in the ratchet speed for hybridogenetic populations. this advantage originates from the model properties that only half of a hybridogen's new mutations occur on its clonally transmitted part and, thus, can be transmitted to the next generation, whereas in an asexual organism all novel mutations are transmitted to the offspring. in this model, fixation of mutations followed the loss of the least loaded fitness classes in asexual populations and hybridogenetic populations. although population size had little influence on the mutational load and on the number of fixed alleles in the size-limited populations with epistatic mutation interaction, hybridogenetically reproducing populations were generally less prone to the accumulation of deleterious mutations than asexual populations. the selective forces against deleterious mutations seemed to be sufficiently effective to prevent a substantial drop in average population fitness, as the mutation rate on the clonally transmitted genome is reduced relative to that on the sexual genome. even in quite small populations, sexually reproducing organisms are at a disadvantage compared to all female asexual populations if the male to female mutation rate ratio is high, at least under quadratic and truncation selection. this result could be due to our choice of the low rate of deleterious mutations per female and generation in the asexual populations. however, the effective values of u are still debated, and the value chosen for our simulations seems to be somewhere in the middle of the reported ranges. furthermore, our u-value of allows a direct comparison with the results of redfield. at high levels of α, small hybridogenetic populations accumulate fewer mutations than small populations of their sexual parent species if mutations interact synergistically. since hybridogenetic populations are better protected against the accumulation and fixation of deleterious mutations than asexual populations, and at the same time, are less affected by the negative impacts of a higher deleterious mutation rates in males, hybridogenetically reproducing organisms perform better than asexual and sexual organisms at reduced population sizes under quadratic and truncation selection if α > this result is quite interesting as this is the area, where the average u is between and, and, therefore, in the range where sexual reproduction becomes favorable over asexual reproduction in mutational deterministic models.  the simulation results in this work indicate that with male to female mutation rates higher than one, hybridogenetic populations can be less prone to the accumulation of deleterious mutations, and hence have a higher population fitness than their sexual counterpart. as α becomes larger, mutation accumulation in the part of the genome derived form the sexual species is faster, and these mutations make it more difficult for new mutations to accumulate on the clonal part of the genome. this effect becomes more pronounced with synergistic forms of epistasis. furthermore, in cases with epistasis and α = although sexual populations have a higher fitness than the hybrids, the hybrids in turn do better than the asexuals, because mutations are purged more effectively in the sexually derived part of their genome. these results lead to the question why hybridogenetic organisms are so scarce compared to sexual species. at the genetic level, one reason could be that, in the real world, mutation interactions are far more complicated and diverse than the ones used in this model. if we assume that mutation interactions vary from negative through no to positive epistasis, with an average effect of no epistasis, sexual reproduction still acts best against the accumulation of mutations in very small populations. also, in natural populations, neither selection nor dominance coefficients are constant values for all deleterious mutations. another reason why hybridogens are scarce could be that the real obstacle is actually to achieve hybridogenetic reproduction. in order to do so, meiosis has to be circumvented or disrupted which often leads to infertility or sterility. furthermore, one cannot consider the evolution of hybridogenesis by only looking at genetic mechanisms but has to include ecology as well. the survival of the hybrid population totally depends on the continued contact with the sexual parental population which serves as an indispensable sperm donor. their sperm-dependence prohibits the hybridogens to occupy niches which are markedly different from the niche of the sexual parent and does not allow out-competing the parental species and leading it to extinction. hence, hybridogenesis can only evolve and persist under ecological conditions that allow stable populations of both, the sexual parasite and their sexual hosts. this ecological issue was not considered in the present work, but has been dealt with in previous publications our result, that clonal or hemiclonal polymorphism can not be maintained with the used model framework, suggests that the often observed diversity in sympatric clones or hemiclones either originates from subsequent recruitments of new clones or hemiclones through repeated primary hybridization, or that diversity in clones is maintained through the occupancy of different microhabitats or "frozen niches" although mutation accumulation is less of a threat to hybridogenetic populations under several of the conditions used in this model, there are other factors inherent to asexual or hybridogenetic reproduction which affect the long time success of such populations: the response to rapidly changing environments or to parasites is still best ensured through sexual reproduction. but nevertheless, the results of this study indicate that, as suggested by schultz, hybridogenetic species could in fact act as a transition state in the formation of new sexual species once recombination is reestablished and reproductive isolation from both sexual ancestors occurred. the surprisingly small accumulation of deleterious mutations on the clonally transmitted part of a hybridogen's genome under synergistic epistasis and differential sex specific mutation rates supports this as an option for new species formations. 
if mutation rates are lower in female gametes than males, it raises the possibility for lower mutation accumulation in a hybridogenetic population, and consequently, higher population fitness than its sexual counterpart. we show through monte-carlo simulations that at higher male to female mutation ratios, and sufficiently large population sizes, hybridogenetic populations can carry a lower mutation load than sexual species. since mutation accumulation can be less pronounced in hybridogenetic populations, the question arises why hybridogenetic organisms are so scarce compared to sexual species. yet, lower mutation accumulation in hybridogenetic populations opens the possibility that hybridogenetic species can develop into new sexual species once recombination is re-established and reproductive isolation from sexual ancestors has occurred. other problems may involve too little genetic diversity to respond to changing environments and problems in becoming hybridogenetic. instead, the paternal genome is discarded from the germline prior to meiosis, and gametes contain solely unrecombined maternal genomes. mutations accumulate faster on the sexual part of the genome, and with the purifying effects of epistasis, it makes it more difficult for mutations to be transmitted on the clonal part of the genome. in contrast, recombining sexual species propagate an intermixed pool of mutations derived from the maternal and paternal parts of the genome. in smaller populations, the same mechanism reduces the speed of muller's ratchet and the number of fixed mutations compared to similar asexual species. they are of hybrid origin but show no recombination between the genomes of their parental species. hybridity is restored each generation by backcrossing with males of the sexual parental species whose genome was eliminated. despite competition with the sexual parental species, hybrid populations are dependent on the maintenance of – and contact with – their sexual counterpart. female only unisexual vertebrates that reproduce by hybridogenesis show an unusual genetic composition. this effect is more pronounced with synergistic forms of epistasis. in considering this, it is likely that comparison of population fitnesses is not sufficient.
 female only unisexual vertebrates that reproduce by hybridogenesis show an unusual genetic composition. they are of hybrid origin but show no recombination between the genomes of their parental species. instead, the paternal genome is discarded from the germline prior to meiosis, and gametes contain solely unrecombined maternal genomes. hence hybridogens only transmit maternally inherited mutations. hybridity is restored each generation by backcrossing with males of the sexual parental species whose genome was eliminated. in contrast, recombining sexual species propagate an intermixed pool of mutations derived from the maternal and paternal parts of the genome. if mutation rates are lower in female gametes than males, it raises the possibility for lower mutation accumulation in a hybridogenetic population, and consequently, higher population fitness than its sexual counterpart.  we show through monte-carlo simulations that at higher male to female mutation ratios, and sufficiently large population sizes, hybridogenetic populations can carry a lower mutation load than sexual species. this effect is more pronounced with synergistic forms of epistasis. mutations accumulate faster on the sexual part of the genome, and with the purifying effects of epistasis, it makes it more difficult for mutations to be transmitted on the clonal part of the genome. in smaller populations, the same mechanism reduces the speed of muller's ratchet and the number of fixed mutations compared to similar asexual species.  since mutation accumulation can be less pronounced in hybridogenetic populations, the question arises why hybridogenetic organisms are so scarce compared to sexual species. in considering this, it is likely that comparison of population fitnesses is not sufficient. despite competition with the sexual parental species, hybrid populations are dependent on the maintenance of – and contact with – their sexual counterpart. other problems may involve too little genetic diversity to respond to changing environments and problems in becoming hybridogenetic. yet, lower mutation accumulation in hybridogenetic populations opens the possibility that hybridogenetic species can develop into new sexual species once recombination is re-established and reproductive isolation from sexual ancestors has occurred. 