The role of deleterious mutations in the stability of hybridogenetic water frog complexes
 Pasquale Bove^{1},
 Paolo Milazzo^{1} and
 Roberto Barbuti^{1, 2}Email author
DOI: 10.1186/1471214814107
© Bove et al.; licensee BioMed Central Ltd. 2014
Received: 28 February 2014
Accepted: 9 May 2014
Published: 16 May 2014
Abstract
Background
Some species of water frogs originated from hybridization between different species. Such hybrid populations have a particular reproduction system called hybridogenesis. In this paper we consider the two species Pelophylax ridibundus and Pelophylax lessonae, and their hybrids Pelophylax esculentus. P. lessonae and P. esculentus form stable complexes (LE complexes) in which P. esculentus are hemiclonal. In LE complexes all the transmitted genomes by P. esculentus carry deleterious mutations which are lethal in homozygosity.
Results
We analyze, by means of an individual based computational model, LE complexes. The results of simulations based on the model show that, by eliminating deleterious mutations, LE complexes collapse. In addition, simulations show that particular female preferences can contribute to the diffusion of deleterious mutations among all P. esculentus frogs. Finally, simulations show how LE complexes react to the introduction of translocated P. ridibundus.
Conclusions
The conclusions are the following: (i) deleterious mutations (combined with sexual preferences) strongly contribute to the stability of LE complexes; (ii) female sexual choice can contribute to the diffusion of deleterious mutations; and (iii) the introduction of P. ridibundus can destabilize LE complexes.
Keywords
Hybridogenesis Water frogs Sexual selection Computational models SimulationsBackground
Lake frog (Pelophylax ridibundus Pallas, 1771) and pool frog (Pelophylax lessonae Camerano, 1882) can mate producing the hybrid edible frog (Pelophylax esculentus Linneus, 1758). P. esculentus can coexist with one or both of the parental species giving rise to mixed populations. Usually, the genotypes of P. ridibundus, P. lessonae and P. esculentus are indicated by RR, LL, and LR, respectively. In Europe there are mixed populations containing P. ridibundus and P. esculentus individuals, called RE systems, populations with P. lessonae and P. esculentus individuals, called LE systems, and populations with all three species. Due to the eastern origin of P. ridibundus, RE complexes are frequently found in Eastern Europe, while LE systems are widespread throughout the rest of Europe [1–5]. Hybrids in these populations reproduce in a particular way, called hybridogenesis[1, 6–12]. Hybridogenesis consists of a gametogenetic process in which the hybrids exclude one of their parental genomes premeiotically, and transmit the other genome, clonally, to eggs and sperm. For example, in LE complexes, P. esculentus hybrids have both the genomes of the parental species, L and R, but they produce only R gametes.
This mode of reproduction requires hybrids to live sympatrically with the parental species whose genome has been eliminated. In this way hybrids in a LE system eliminate the L genome thus producing P. esculentus when mating with P. lessonae, and generating P. ridibundus when mating with other hybrids. Usually P. ridibundus generated in LE complexes are inviable due to deleterious mutations accumulating in the clonally transmitted R genome [13–17]. Analogously, in RE systems there is a tendency during hybrid gametogenesis to eliminate the R genome; as with LE systems, P. lessonae, the offspring of hybrids, are often inviable.
In natural LE complexes, the inviability of offspring of P. esculentus ×P. esculentus matings is evidenced by the absence of adults P. ridibundus. Experimental crosses between coexisting hybrids (from localities sampled throughout the range of LE populations) also show that such offspring are inviable [6, 14, 16, 17]. These studies have also revealed that the same hybrid individuals, producing inviable progeny, produce viable progeny when crossed with either parental species or with hybrids from different regions. The lethality of natural hybrid × hybrid matings is thus neither the result of hybrid sterility nor the inherent consequence of the hemiclonal reproductive mode. Guex et al. present two simple hypotheses, both explaining the observed inviability of P. esculentus ×P. esculentus progeny by the load of deleterious mutations on the clonally transmitted R genomes [16]. Quoting from their paper, the hypotheses are “(1) inviability is caused by homozygosity for recessive deleterious mutations at particular gene loci; or (2) inviability is caused by a general deterioration of nonrecombining R genomes through Muller’s ratchet, reflecting different hemiclonespecific sets of incompletely recessive mutations, which leads to lethality when two such deteriorated R genomes are combined”. Their conclusion is that the hypotheses are not mutually exclusive, however there is evidence to support the plausibility of the first hypothesis: Muller’s ratchet generates deleterious mutations in relatively random places in the genome, which are then likely to be different in different geographical areas. The study in [16] suggests that, in some cases, single lethal mutations may have the same effect as the accumulation of deleterious ones. However, most studies on P. esculentus fitness, suggest that when in a heterozygous state the effect of deleterious mutation is not significant.
Reproductive pattern of water frogs
LL  LR  

L _{ y } L  L _{ y } L L L  L _{ y } R L R 
L _{ y } R  L R  RR not viable 
The Y chromosome determines the sex of frog males and can occur only in the L genome, due to primary hybridization which involves, due to size constraints, P. lessonae males and P. ridibundus females. Only one of the three possible matings resulting in viable offspring produces LL genotypes (Table 1). This would give an advantage to P. esculentus which could outnumber P. lessonae and eventually eliminate them. This situation would also eventually result in an extinction of P. esculentus which cannot survive without the parental species. In addition to their relative abundance which is promoted by the above reproductive pattern, P. esculentus have other advantages. Although in many cases they show either no differences or intermediate characteristics compared to their parental species [24–27], P. esculentus show behavioural differences [28, 29], and, have, by heterosis, a greater fitness than the parental species in certain aspects [13, 15, 30–36]. The combination of the relative abundance and heterosis should outcompete P. lessonae in LE complexes. The widespread distribution of LE complexes, although with different percentages of hybrids, reveals that there are mechanisms which contribute to the stability of such complexes [37–40]. Of these mechanisms, sexual selection seems to be one of the most important. In fact, P. esculentus females prefer (either overtly or cryptically) P. lessonae males than males of their own species [41–45]. Many mathematical and computational models have studied the influence of sexual selection in the evolution of populations [46–54]. In addition some models have focused on sexual selection in complexes in which some form of clonal reproduction exists [55–57]. The models in [57–59] show how female preference is able to stabilize LE complexes by counterbalancing both heterosis and the reproductive advantage of P. esculentus. Other factors, such as reproductive performance, in conjunction with sexual choice can increase the stability of LE complexes [60].
Using an individualbased computational model, in this paper we study three problems. The first is how deleterious mutations contribute to the stability of LE complexes. The second concerns how, in LE complexes, deleterious mutations can diffuse in the R genomes of the whole P. esculentus population. The third is the invasiveness of P. ridibundus in LE complexes.
Regarding the first problem, the aim is to investigate whether deleterious mutations on the R genome contribute, together with female preferences, to the stability of LE complexes.
As for the diffusion of deleterious mutations in the population, an interesting hypothesis is proposed in [61], in which deleterious mutations can influence female preferences. From the literature we know that females of P. esculentus have a strong preference for P. lessonae males. Vorburger et al. in [61] suggest that, of P. esculentus males, P. esculentus females may prefer those with mutations on the R genome. Such mutations could make the affected loci on the R genome dysfunctional, thus producing a more “lessonaelike” genotype.
Finally, regarding the third problem, Vorburger and Reyer in [9] suggest that the introduction of P. ridibundus can either provoke the collapse of LE populations, or result in a replacement by P. ridibundus of both P. lessonae and P. esculentus, leading to a monospecific population.

Is the role of deleterious mutations necessary for the stability of LE complexes?

How can a stable LE complex be obtained?

What is the effect of introducing P. ridibundus into LE complexes?
Methods
The model
To study the interaction between populations of P. lessonae, P. esculentus and P. ridibundus we developed an individualbased model. To answer the three questions above, we started with a simple model (for the first question) and then extended it step by step (to tackle the second and third questions).
In the simplest model, we consider diploid individuals whose genotype is represented by two chromosome types: L and R. Chromosomes R can contain deleterious mutations (represented by R_{ d }), and only chromosomes L can have the sexdetermining chromosome Y (represented by L_{ y }). Thus the possible genotypes are: LL, L_{ y }L, LR, L_{ y }R, L R_{ d }, L_{ y }R_{ d }, RR, R R_{ d }, R_{ d }R_{ d }.
where σ is a parameter measuring the strength of the ecological selection (smaller values of σ correspond to a stronger selection). In the simulations we use two different values for σ, σ=0.4, which corresponds to a hard environment, and σ=0.6, which corresponds to a weaker selection.
where δ_{ h } and δ_{ e } describe the fitness decrement associated with homozygous genotypes (which do not gain from heterosis), and P. ridibundus genotypes, respectively. The use of a further decrement, δ_{ e }, in the chromosome fitness of P. ridibundus, derives from the fact that LE complexes usually live in pools and marshes, where P. ridibundus are less fit. In the following sections, we use δ_{ h }=0.2 and δ_{ e } will assume the values 0.0, 0.2, and 0.4. δ_{ e }=0.0 means that P. ridibundus have the same fitness as P. lessonae (i.e. the environment includes niches for both species), while δ_{ e }=0.4 represents the fact that P. ridibundus are strongly disadvantaged compared to P. lessonae (i.e. the environment consists of a typical P. lessonae habitat).
Note that the greater n is, the greater the strength of the female preference. Increasing the n value will lead to a female choosing from a greater number of males, thus mimicking the behaviour of a more discriminating female. In order to obtain stable complexes, we assume a speciesspecific female preference, in particular, following the studies in [59, 60], we assign a stronger preference to females of the parental species than the hybrid females. P. esculentus females have the same behaviour as P. lessonae, thus competing for the same kind of males. Likewise, P. ridibundus females prefer more “ridibunduslike” males. Hereafter we call “lessonae preference” this kind of female preference, because P. lessonae males are the most preferred. The number of n candidates, which each female chooses from, is set to 30, 15, and 30 for the P. lessonae, P. esculentus, and P. ridibundus females, respectively. Choosing from among 30 candidates has a biological flavour. If males are distributed in the environment with a density of one male per square meter, each female must swim in a circle of 10 meters of diameter in order to check out 30 males, a distance which is reasonable for a frog. In [57–59] it is shown that stable populations are only found when the preference of P. lessonae females is greater than the preference of P. esculentus females. In these papers it is shown that, under the above assumption, many different values of female preference lead to stable complexes. We have performed many simulations by leaving the value of n for P. lessonae and P. ridibundus unmodified, and by varying the value of n for P. esculentus. We have found that if the ratio of the value of P. esculentus to the value of the parental species belongs to the interval [ 0.03,0.7] we obtain stable complexes, when P. ridibundus are inviable. This result is analogous to the ones in [57–59]. Because different values in the interval [ 0.03,0.7] affect only the percentage of hybrid frogs in the final stable population, the choice of n does not change the overall dynamics of the population. For this reason we use the nonextreme value 0.5.
Offsprings genotypes are obtained from the gamete combination of the parents.
where b is the average number of offspring for females that can reach the stage of adults, ϕ is the percentage of females in the population, N is the number of individuals in the population competing for the resources, and $\mathcal{K}\left(\phantom{\rule{0.3em}{0ex}}g\right)$ is the carrying capacity associated with the genotype g. $\mathcal{K}\left(\phantom{\rule{0.3em}{0ex}}g\right)$ is given by $\mathcal{F}\left(\phantom{\rule{0.3em}{0ex}}g\right){\mathcal{K}}_{0}$ where ${\mathcal{K}}_{0}$ is the maximum carrying capacity of the environment. In all our simulations we assume b=6 and ${\mathcal{K}}_{0}=3000$. The standard BevertonHolt model is modified due to the fact that we consider overlapping generations and we apply the viability selection based on survival probability, not only to young tadpoles but to all the individuals in the population. The simplest model is used to answer the first question by performing both simulations in which all R genomes carry deleterious mutations and simulations in which all R genomes are free from deleterious mutations.
To answer the second question, “How can a stable LE complex be obtained?”, we simulate the diffusion of deleterious mutations in the population, starting with an LE complex, composed by P. lessonae, in which there are only a few P. esculentus individuals (without mutations on the R genome). We consider a mutation rate, μ, which gives the probability of adding a new deleterious mutation on the R genome of an offspring (in the simulations μ is set both to 10^{4} and to 10^{5}). We consider different “stages” in the accumulation of mutations in the R genome. For the sake of simplicity we consider only three stages: R, R_{d1}, and R_{ d }. R is the genome without mutations, R_{d1} is the genome with a non lethal accumulation of mutations, and, as before, R_{ d } is the final stage of accumulation (lethal in homozygous individuals). With this scenario we have P. ridibundus females with the following possible genotypes (in order of decreasing fitness): RR, R R_{d1}, R R_{ d }, R_{d1}R_{d1}, R_{d1}R_{ d }, R_{ d }R_{ d }, where R_{ d }R_{ d } is the only lethal genotype. The fitness decrease of each genotype with respect to the previous one is given by δ_{ m }. In the simulations we set δ_{ m }=0.04. In this scenario, following [61], we assume that P. lessonae and P. ridibundus females have a strong preference for males of their own species, while of P. esculentus males, females prefer males with a more “lessonaelike” genotype (males with mutations on the R genotype). We call this kind of female preference “lessonaelike preference”. We have P. esculentus males with the following genotype: L_{ y }R, L_{ y }R_{d1}, L_{ y }R_{ d }, which, according to the “lessonaelike preference”, are in order of increasing preference by the female.
Finally, to answer the third question, “What is the effect of introducing P. ridibundus into LE complexes?”, we need to simulate the effect of the introduction of P. ridibundus males and females in an LE complex. In these simulations we consider both R genomes without mutations; this is because we assume the absence of the Muller ratchet in the (sexually reproducing) introduced P. ridibundus and R_{ d } genomes generated by the stable LE complex. The further extension we consider is the possibility of having P. ridibundus males, i.e. the possibility of having the Y chromosome on R genomes as well: R_{ y } and R_{ y d }.
We introduced a limit for the lifespan of individuals, thus all individuals exceeding 10 years of age are removed from the population. Removing old frogs from the system avoids that extremely fit individuals survive indefinitely because the viability selection is not able to remove them. This situation can easily happen in models with overlapping generations which do not consider deaths due to aging.
Parameters used in the model
σ  Selection strength  0.40.6 
δ _{ h }  Fitness decrease due to homozygosity  0.2 
δ _{ e }  Fitness decrease for environmental reasons  0.00.4 
δ _{ m }  Fitness decrease for each step of mutation accumulation  0.04 
n  Number of attempts per female in choosing males, P. lessonae, P. esculentus, P. ridibundus  30, 15, 30 
b  Number of surviving tadpoles for each female  6 
${\mathcal{K}}_{0}$  Environmental carrying capacity  3000 
μ  mutation rate  10^{5}, 10^{4} 
Implementation
The process was formalize with CLS [69–75] and then implemented in C++ language using Microsoft Visual Studio 2010. Each simulation runs for 2000 iterations (more when necessary) with a carrying capacity of 3000 individuals. A single simulation takes nearly 10sec on a worstation equipped with Intel i5 3.2Ghz processor and 8GB RAM.
Results
Is the role of deleterious mutations crucial for the stability of LE complexes?
The result of these simulations is not surprising. Essentially the results are in accordance with those in [58, 59], showing that female preference is a strong stabilizing force for LE complexes.
Reproductive pattern of water frogs without deleterious mutations
LL  LR  RR  

L _{ y } L  L _{ y } L L L  L _{ y } R L R  L _{ y } R L R 
L _{ y } R  LR  RR  RR 
Note that the higher the percentage of P. esculentus in the initial population, the smaller the number of generations before the collapse. By varying either the strength of female preferences or the strength of ecological selection, we obtain different values for the number of generations before the collapse, but an evolution towards collapse remains the overall trend of the system. Even when P. ridibundus are at a serious disadvantage, δ_{ e }=0.4, RR females are able to compromise the stability of LE complexes. Of course, we amplify this effect by decreasing δ_{ e }, thus we do not show the population dynamics with δ_{ e }=0.0. In this case, the point of collapse is reached very quickly.
How can a stable LE complex be obtained?
In this section we study the effect of both “lessonae preference” and “lessonaelike preference” of P. esculentus and P. lessonae females in the diffusion of deleterious mutations. We set the mutation rate, μ, either to 10^{4} or to 10^{5}. We consider three possible stages of deleterious mutation accumulation in each R genome (R, R_{d1}, and R_{ d }), thus any mutation event determines the passage from one stage to the next. We start the simulations with two different initial populations. The consistency of the initial P. lessonae population is the same in all the simulations, 2700 individuals, but the number of P. esculentus individuals is set, initially, either to 10 or 100 individuals. We also perform simulations with two values for δ_{ e }, which lead to a decrease in fitness of mutationfree P. ridibundus compared to P. lessonae. We set δ_{ e } to either 0.2 or 0.4, i.e. we consider that the environment is either weakly penalizing for P. ridibundus or is a typical P. lessonae habitat, which is not suitable for P. ridibundus frogs. In addition, any further mutation accumulation on the R genomes of P. ridibundus decreases their fitness by δ_{ m }=0.4.
Outcomes of 100 simulations with an initial P. esculentus population of 10 and 100 individuals without deleterious mutations on the R genome
a)δ_{ e }=0.2  

Initial P. esculentus pop.  10  100 
Stable LE system  0  0 
LL population  0  0 
Collapse  100  100 
b) δ _{ e } =0 . 4  
Initial P. esculentus pop.  10  100 
Stable LE system  0  0 
LL population  0  0 
Collapse  100  100 
Outcomes of 100 simulations with an initial P. esculentus population of 10 and 100 individuals without deleterious mutations on the R genome
a)δ_{ e }=0.2  

Initial P. esculentus pop.  10  100 
Stable LE system  0  0 
LL population  0  0 
Collapse  100  100 
b) δ _{ e } =0 . 4  
Initial P. esculentus pop.  10  100 
Stable LE system  76  73 
LL population  1  0 
Collapse  23  27 
Outcomes of 100 simulations in which in a stable LE complex a percentage of 5% of P. ridibundus , males and females, are introduced
δ_{ e }=0.0  δ_{ e }=0.2  δ_{ e }=0.4  

Stable LE system  0  46  100 
Stable LER system  32  0  0 
Stable P. ridibundus population  39  0  0 
Collapse  29  54  0 
What is the effect of the introduction of P. ridibundusin LE complexes?
In this section we analyze the effect of introducing P. ridibundus into an LE complex. This scenario really happens in natural environments due to the importation in Western Europe of P. ridibundus for commercial purposes.
In order to study the effect of the translocation of P. ridibundus, we performed simulations by varying the fitness of the introduced frogs (δ_{ e }=0.0, 0.2, 0.4). δ_{ e }=0.0 means that the environment does not put P. ridibundus at a disadvantage with respect to P. lessonae. In these simulations we consider a strong selection strength, σ=0.4.
If the fitness of P. ridibundus is equal to the fitness of P. lessonae (δ_{ e }=0.0), in most cases the population becomes a monospecific population of P. ridibundus. Note that there are many cases in which the three species coexist at the end of the simulations, which we discuss explicitly in the following section. If the fitness of P. ridibundus decreases (δ_{ e }=0.2), the introduced frogs do not survive for long. However, before their extinction P. ridibundus can mate with P. lessonae and P. esculentus, thus introducing mutationfree R genomes into the hybrid population. At this point, viable P. ridibundus females born from matings between hybrids lead the population to collapse. Finally, when the fitness of P. ridibundus is very low (δ_{ e }=0.4), P. ridibundus frogs are immediately expelled from the system.
Discussion
From the results of the previous section we can deduce the overall dynamics of LE complexes with viable P. ridibundus females. In general, the presence of viable P. ridibundus females significantly changes the reproductive outcome of LE complexes, and the generation of offspring becomes as depicted in Table 3. For the sake of simplicity let us consider hypothetical complexes in which the number of females is the same for the three species. For such populations the production of P. lessonae offspring passes from 33.33% in stable LE complexes (recall that in such complexes P. ridibundus are inviable and do not survive) to 16.66% in complexes where P. ridibundus survive. Thus the viability of P. ridibundus decreases the relative abundance of P. lessonae offspring. The decrease in the relative production of P. lessonae offspring leads to a decrease in P. lessonae adults in the future. This, in turn, causes a decrease in the production of L gametes, which are produced only by P. lessonae frogs. This process, if not stopped by some external reasons, results in a trend towards the extinction of P. lessonae. A population composed only of P. esculentus individuals and P. ridibundus females cannot survive, because no gametes with the Y chromosome can be generated.
The discussion which follows, regarding the three questions mentioned previously, is based on the trend towards extinction mentioned above, modulated for example by the accumulation of mutations, or the introduction of viable P. ridibundus males.
Deleterious mutations are necessary for the stability of LE complexes
We showed in Section ‘Is the role of deleterious mutations crucial for the stability of LE complexes?’ that by using the “lessonae preference”, we essentially obtain the same results as [58, 59]. If there are no viable P. ridibundus offspring, the system evolves towards stability regardless of the strength of the selection. The same complexes, with the same values of female preferences, but without deleterious mutations on R genomes will collapse irrespectively both of the selection strength and the fitness of viable P. ridibundus. In the dynamics of the population towards collapse, there are roughly three phases. In the first phase, the effect of sexual selection and the abundance of P. lessonae males maintains a low number of P. ridibundus offspring. Although no mutations are present in the P. esculentus genome, a very few P. ridibundus females are generated because mating between hybrids is rare, due to female preferences. In the second phase, the number of hybrids increases because of their fitness as a result of heterosis, while the number of P. lessonae decreases. In this phase the increased number of hybrids facilitates mating which thereby produces viable P. ridibundus females. In the third phase, P. ridibundus females act as sexual parasites for hybrids, and their number grows because of P. esculentus ×P. esculentus and P. esculentus ×P. ridibundus matings. Note that these matings produce only P. ridibundus females. Viable P. ridibundus females will mate, preferably, with P. esculentus males (which have a more “ridibundus phenotype”) producing a bigger population of ridibundus females for any further generation. In the model we do not consider male frogs as a limiting factor for reproduction, that is one male is able to fertilize the eggs of a unlimited number of females. We consider also that males will not make any specific choice of females [45]. The number of P. esculentus decreases because of the reduced number of P. lessonae. This phase ends with a population of all P. ridibundus females which quickly collapses (Figure 2).
Consequently, if stability is maintained by female preferences, collapse only can be prevented if the initial population of P. esculentus is affected by deleterious mutations on the R genomes, which prevents the birth of viable P. ridibundus. Thus we can conclude that, in a system in which the parameters used for the stability are essentially female preferences and viability of P. ridibundus, deleterious mutations are necessary for the stability of the complex. These results help to clarify why natural LE complexes generating viable P. ridibundus are extremely rare, and in most cases the percentage of viable P. ridibundus is not significant [15]. It is difficult (perhaps impossible) for LE complexes to persist if they generate viable P. ridibundus. Starting with an initial population in which all P. esculentus frogs carry deleterious mutations, the complex evolves towards a stable configuration (Figure 1). The possible collapse of the population occurs only if P. esculentus outcompete P. lessonae, however this evolution seldom takes place because of the large number of P. lessonae in the initial population. The results show that, with the same selection strength, LE complexes evolve towards the same percentages of the two frog species, whatever the initial percentages.
Our results regarding the stability of LE complexes both with deleterious mutations in all R genomes and with a “lessonae preference” are similar to those in [57–59]. We show that deleterious mutations strongly influence the stability of LE complexes. Thus deleterious mutations are not only a secondary consequence of Muller’s ratchet but they have an important role in the stability of complexes. In this paper we highlight that, if we consider both heterosis and females preferences as the only stabilizing forces of LE complexes, the lack of deleterious mutations drives such populations to collapse. Thus neither female preferences nor deleterious mutations are sufficient to maintain the stability of LE complexes however, in this scenario, each is necessary for stability.
Female preference can contribute to obtaining stable LE complexes
The above discussion highlights the important role of deleterious mutations in stable LE complexes. But how can stable LE complexes be obtained?
Our results show that, in order to reach stable LE populations, there must be forces that drive the diffusion of deleterious mutations. Even with a fast mutation rate, 10^{4}, Muller’s ratchet alone is not sufficient for diffusing the deleterious mutations in the population. The “lessonae preference” does not allow P. lessonae and P. esculentus females to choose from among P. esculentus males. Thus the diffusion of deleterious mutation among P. esculentus individuals and P. ridibundus females is not guided by female preferences, but only by the mutation rate (essentially by Muller’s ratchet). In all our simulations this diffusion turns out to be very slow, thus viable P. ridibundus females are generated before Muller’s ratchet accumulates lethal mutations on all R genomes. This intermediate phase, with a sufficient number of viable P. ridibundus females, is responsible for the collapse of the whole system.
Selecting R genomes with a higher mutation accumulation is accelerated by the “lessonaelike preference”. In this case the production of offspring with greater mutation accumulation on the R genome is favoured, and consequently the production of fit P. ridibundus females is lowered. To understand this process we need to consider that mutation accumulation on R genomes decreases the fitness of P. ridibundus females, but it does not affect the fitness of P. esculentus in which dysfunctional R genomes are counterbalanced by “healthy” L genomes.
Another important point is that a significant parameter in the diffusion of deleterious mutations is the fitness of P. ridibundus to the environment. If this fitness is too high, too many viable P. ridibundus females are produced before significant mutation accumulations. We know that a high number of such viable females will lead the population to collapse.
For computational purposes we have considered only three stages of mutation accumulation on the R genomes. This is an approximation of the Muller’s ratchet which, in most cases, operates through a huge number of mutations. We approximate the Muller’s ratchet by decreasing the mutation rate so that a mutation in the model corresponds to many mutations in real genomes. Following the estimation in [76], we assume that, in an eukaryote organism, the mutation rate in a whole genome during sexual reproduction is in the interval [3×10^{2},9×10^{1}]. Many of these mutations are either not significant or not deleterious. Values of the mutation rate in the interval [10^{5},10^{4}], used in our model, take both the above considerations into account.
Our study differs significantly with the results of other authors with regard to the diffusion of mutations. The models in [58–60] provide an extensive insight into the reasons for the stability of LE complexes starting from the assumption that deleterious mutations are present in all the R genomes in the population. Our model builds on the previous ones by assuming sexual choices in the populations. However, it differs by considering a population in which deleterious mutations are not present, but are generated according to a mutation rate. In addition, only when this accumulation reaches a given threshold does it become lethal. This leads us to conclude that sexual selection not only stabilizes the complexes, but can also force mutation diffusion.
Note that our simulations do not enable us to prove the hypothesis suggested in [61]. Computational and mathematical models, without subsequent experimental support, can only be used to rule out incorrect hypotheses  they cannot prove correct ones. Computational and mathematical models can only state that a hypothesis is plausible. In the case of LE complexes, the stabilization period is so long that no real experiment can support a hypothesis on its stabilization, however our model suggests that the hypothesis in [61] regarding female preference, could plausibly lead to stable LE complexes.
Invasion of translocated P. ridibundus
Another point that we study with our model is the consequence of introducing P. ridibundus into stable LE complexes. P. ridibundus can mate both with P. esculentus, producing P. ridibundus, and with P. lessonae (primary hybridization), producing P. esculentus. Primary hybrids can have low fertility rates [77], thus their contribution to the dynamics of the population is low. In our model we take account of this low contribution by decreasing the possibility of producing primary hybrids. This is done by setting the female preferences of P. ridibundus in a way that P. lessonae males are seldom chosen (the value of n, in the bestof n procedure, is set to 30). On the other hand, the preference of P. lessonae females is mainly for males of their own species. We assign the same fitness both to the introduced P. ridibundus and to those generated by matings of P. ridibundus with P. esculentus. This is in accordance with the seminatural experiments in [17]. However, in some simulations we use a P. ridibundus fitness, which is lower than the fitness of the resident P. lessonae and P. esculentus because we consider that a marshy environment with a low oxygen level, where P. lessonae and P. esculentus live, is less suitable for P. ridibundus.
The results show that, as predicted in [9], the introduced P. ridibundus often outcompete the other species resulting in a monospecific population, when their fitness is comparable to the resident population’s. Although the introduction of P. ridibundus results in new R hemiclones, which contribute to the genetic diversity of hybrids, our results do not support the hypothesis presented in [5] that this genetic diversity can stabilize the hybridogenetic system. If the fitness of P. ridibundus is competitive with the fitness of the resident population (δ_{ e }=0.0), P. ridibundus males will survive and P. ridibundus often will replace the original population. Note that in this case we also have stable LER complexes as the outcome. This is a system where two independent populations coexist, an LE complex and a P. ridibundus population. The LE complex is stable due to female preferences and lethal mutations on the R genomes, while the P. ridibundus population is stable because of the absence of deleterious mutations, which are purged by selection. The two populations do not cross because the number of individuals in both is high enough to ensure that females of one population in most cases find a preferred male, of the same population, in the set randomly chosen by the bestof n procedure.
The whole population collapses when the introduced frogs have a low fitness. In this case, P. ridibundus individuals will not survive for long, given their unfitness, but P. ridibundus frogs, before their death, can introduce R genomes without mutation in the P. esculentus population, thus provoking the collapse of the complex.
Finally, by assuming that P. ridibundus are at a considerable disadvantage, the introduced unfit population is outcompeted. During their short survival time, P. ridibundus females are not able to have a sufficient number of matings with P. esculentus males, thus they cannot introduce a sufficient number of R genomes without mutations into the P. esculentus population.
Conclusions
We have presented an individualbased computational model to study LE water frog populations, i.e. complexes composed of P. lessonae and P. esculentus. The individual based model considers not only the genotypes, but also the age of each individual and the average lifespan. In addition, female preferences (implemented by a bestof n procedure) and ecological selection are considered.

deleterious mutations in the R genomes strongly contribute, together with sexual preferences, to maintaining the stability of such complexes,

female sexual choice can contribute to the diffusion of R genomes carrying deleterious mutations in LE complexes, and

the introduction of P. ridibundus can destroy LE complexes.
Declarations
Acknowledgements
This study has been supported by the University of Pisa. We thank the anonymous referees for their suggestions that allowed us to improve the paper.
Authors’ Affiliations
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