Abstract
Many parthenogenetically reproducing animals produce offspring not clonally but through different mechanisms collectively referred to as automixis. Here, meiosis proceeds normally but is followed by a fusion of meiotic products that restores diploidy. This mechanism typically leads to a reduction in heterozygosity among the offspring compared to the mother. Following a derivation of the rate at which heterozygosity is lost at one and two loci, depending on the number of crossovers between loci and centromere, a number of models are developed to gain a better understanding of basic evolutionary processes in automictic populations. Analytical results are obtained for the expected neutral genetic variation, effective population size, mutation–selection balance, selection with overdominance, the spread of beneficial mutations, and selection on crossover rates. These results are complemented by numerical investigations elucidating how associative overdominance (two off-phase deleterious mutations at linked loci behaving like an overdominant locus) can in some cases maintain heterozygosity for prolonged times, and how clonal interference affects adaptation in automictic populations. These results suggest that although automictic populations are expected to suffer from the lack of gene shuffling with other individuals, they are nevertheless, in some respects, superior to both clonal and outbreeding sexual populations in the way they respond to beneficial and deleterious mutations. Implications for related genetic systems such as intratetrad mating, clonal reproduction, selfing, as well as different forms of mixed sexual and automictic reproduction are discussed.
- automixis
- parthenogenesis
- neutral genetic variation
- overdominance
- mutation–selection balance
- central fusion
THE vast majority of animals and plants reproduce via the familiar mechanism of sex (Bell 1982): haploid gametes are produced through meiosis, and these fuse to form diploid offspring that are a genetic mix of their parents. Conversely, bacteria, many unicellular, and some multicellular eukaryotes reproduce clonally, i.e., their offspring are genetically identical to their mother. These two extreme genetic systems can also be alternated, e.g., a few generations of clonal reproduction followed by one round of sexual reproduction. Such systems are found in many fungi (e.g., yeast) but also in animals such as aphids that exhibit “cyclical parthenogenesis.” However, there are also genetic systems that resist an easy classification into “sexual” and “asexual.” Among them are automixis and related systems in which a modified meiosis takes place in females, leading to offspring that develop from unfertilized but diploid eggs and that may be genetically diverse and distinct from their mother (Suomalainen et al. 1987; Stenberg and Saura 2009). Explicably, there is much confusion and controversy about terminology in such systems, with some authors referring to them as asexual (because there is no genetic mixing between different lineages) and others as sexual (e.g., because they involve a form of meiosis and/or resemble selfing). Without entering this debate, I will adopt the former convention here, acknowledging that the latter is also valid and useful in some contexts. Also note that clonal, “ameiotic” reproduction in animals is usually referred to as apomixis but that this term has a different meaning in plants (Asker and Jerling 1992; van Dijk 2009).
A good starting point for understanding automixis is to consider a specific system. One system that is particularly well studied is the South African honeybee subspecies Apis mellifera capensis, the Cape honeybee (reviewed in Goudie and Oldroyd 2014). Within A. mellifera capensis, workers can lay unfertilized eggs that develop parthenogenetically into diploid female offspring via a mechanism called “central fusion” (Verma and Ruttner 1983; Suomalainen et al. 1987). Here, meiosis proceeds normally, producing four haploid nuclei, but diploidy is then restored through fusion of the egg pronucleus with the polar body separated in meiosis I. This means that in the absence of recombination between a given locus and its associated centromere, the maternal allelic state at this locus is restored and in particular, heterozygosity is maintained. However, crossover events between a locus and its centromere can erode maternal heterozygosity, leading to offspring that are homozygous for one allele (see below for details and Figure 1). Although workers can produce diploid female offspring asexually, queens (which may be the daughters of workers) still mate and reproduce sexually. However, this system has also given rise to at least three lineages (two historical and one contemporaneous) that reproduce exclusively through central fusion automixis and parasitize colonies of another, sexual honeybee subspecies (A. mellifera scutellata). The contemporaneous lineage (colloquially referred to as the “clone” in the literature) appeared in 1990 and has been spreading rapidly since, causing the collapse of commercial A. mellifera scutellata colonies in South Africa (the “Capensis Calamity”; Allsopp 1992). Heterozygosity levels are surprisingly high in this lineage given its mode of automictic reproduction (Baudry et al. 2004; Oldroyd et al. 2011). Initially, it was hypothesized that this is due to suppression of recombination (Moritz and Haberl 1994; Baudry et al. 2004), as this would make central fusion automixis akin to clonal reproduction. However, more recent work indicates that it is more likely that natural selection actively maintains heterozygosity (Goudie et al. 2012, 2014).
Illustration of the genetic consequences of automixis at a single locus. (A) Starting from a heterozygous mother, during prophase I, crossover events between the locus can induce switches between two possible states. Each crossover converts the original state where identical alleles are linked to the same centromere to the opposite state where different alleles are linked to one centromere. With probability 1/2, crossovers can then revert the second to the first state. (B) Depending on which state is reached, meiosis will result in one of two possible genetic configurations. (C) Fusion of meiotic products or suppression of the first mitotic division can then lead to different proportions of zygote genotypes.
Several other species also reproduce exclusively or facultatively through central fusion automixis, including other hymenopterans (e.g., Beukeboom and Pijnacker 2000; Belshaw and Quicke 2003; Pearcy et al. 2006; Rabeling and Kronauer 2013; Oxley et al. 2014), some dipterans (Stalker 1954, 1956; Murdy and Carson 1959; Markow 2013), moths (Seiler 1960; Suomalainen et al. 1987), crustaceans (Nougue et al. 2015), and nematodes (Van der Beek et al. 1998). Another mechanism of automictic parthenogenesis is terminal fusion. Here, the egg pronucleus fuses with its sister nucleus in the second-division polar body to form the zygote. With this mechanism, offspring from heterozygous mothers will become homozygous for either allele in the absence of recombination, but may retain maternal heterozygosity when there is recombination between locus and centromere. Terminal fusion automixis has been reported, for example, in mayflies (Sekine and Tojo 2010), termites (Matsuura et al. 2004), and oribatid mites (Heethoff et al. 2009). [Note, however, that in mites with terminal fusion automixis, meiosis may be inverted so that the consequences are the same as for central fusion (Wrensch et al. 1994).] Terminal fusion also seems to be the only confirmed mechanism of facultative parthenogenesis in vertebrates (reviewed in Lampert 2008). The most extreme mechanism of automixis is gamete duplication. Here, the egg undergoes either a round of chromosome replication without nuclear division, or a mitosis followed by fusion of the resulting two nuclei. In both cases, the result is a diploid zygote that is completely homozygous at all loci. Gamete duplication has been reported in several groups of arthropods and in particular is frequently induced by inherited bacteria (Wolbachia) in hymenopterans (Stouthamer and Kazmer 1994; Gottlieb et al. 2002; Pannebakker et al. 2004). Finally, there are a number of genetic systems that are cytologically distinct from automixis but genetically equivalent or similar, including intratetrad mating and selfing (see Discussion).
The peculiar mechanisms of automixis raise a number of questions. At the most basic level, one could ask why automixis exists at all. If there is selection for asexual reproduction, why not simply skip meiosis and produce offspring that are identical to their mother? Are there any advantages to automixis compared to clonal reproduction, or are there mechanistic constraints that make it difficult to produce eggs mitotically? How common is automixis, and how can it be detected and distinguished from other modes of reproduction using population genetic methods? What is the eventual fate of populations reproducing via automixis? Are the usual long-term problems faced by populations that forego genetic mixing (such as Muller’s ratchet or clonal interference) compounded in automictic populations because they also suffer from a form of inbreeding depression due to the perpetual loss of heterozygosity, or can the loss of heterozygosity also be beneficial in some circumstances?
Answering these questions requires a firm understanding of how key evolutionary forces such as selection and drift operate in automictic populations. Although a few studies have yielded important insights into this issue, these studies have been limited to specific settings, e.g., dealing with the initial fitness of automictic mutants (Archetti 2004), selective maintenance of heterozygosity (Goudie et al. 2012), or the “contagious” generation of new automictic lineages in the face of conflicts with the sex-determination mechanism (Engelstädter et al. 2011). Here, I develop mathematical models and report a number of analytical and numerical results on the evolutionary genetics in automictic populations. As a foundation, I first (re)derive the expected distribution of offspring genotypes with up to two loci and under different modes of automixis in relation to crossover frequencies. Next, results on several statistics describing neutral genetic diversity are derived. I then investigate how natural selection on deleterious, overdominant, and beneficial mutations operates in automictic populations. Finally, using the previous results, I investigate the evolution of recombination suppression in automicts; a process that in extreme cases might effectively turn automictic into clonal reproduction.
Models and Results
Recombination and loss of heterozygosity
For a single locus, the relationship between crossover rates and loss of heterozygosity during automixis has been previously analyzed by several authors (Pearcy et al. 2006, 2011; Engelstädter et al. 2011), so only a brief summary will be given here. The process can be understood by considering two steps. First, crossover events between the focal locus and its associated centromere during prophase I may or may not produce “recombinant” genotypic configurations following meiosis I prophase (Figure 1A). Second, the resulting configuration (Figure 1B) may then either retain the original heterozygous state or be converted into a homozygous state (Figure 1C).
In step 1, crossovers induce switches between two possible configurations (Figure 1A). A crossover invariably produces a transition from the original state where sister chromatids carry the same allele to the state where sister chromatids carry different alleles, but only produces the reverse transition with probability 1/2. Based on this, it can be shown that with 
crossovers, the probability of arriving in the recombinant state is
(1)If we assume a Poisson distribution of crossover events with a mean of 
we obtain
(2)as the expected fraction of recombinant configurations. Of course, more complex distributions that take crossover interference into account could also be applied to Equation 1 (Svendsen et al. 2015). It can be seen from Equations 1 and 2 that when the number of crossovers increases (i.e., with increasing distance of the locus from the centromere), the expected fraction of recombinant configurations converges to 2/3.
In step 2, the meiotic products form diploid cells through different mechanisms, resulting in three possible genotypes in proportions as shown in Figure 1C. Combining Equation 2 with these probabilities yields the following overall probabilities of conversion from heterozygosity in the mother to homozygosity in her offspring for central fusion (CF), terminal fusion (TF), and gamete duplication (GD) automixis:
(3)These equations can also be expressed in terms of map distance d (in Morgans) between the locus and its centromere, which may be known in sexual conspecifics or related sexual species. This is done simply by replacing 
with 2d. (The factor 2 arises because a single crossover event results in recombination in only half of the gametes produced by a sexually reproducing individual.)
Let us next consider two loci. Offspring proportions when at most one locus is heterozygous can be readily deduced from the single-locus case outlined above. Similarly, when the two loci are on different chromosomes, predicting offspring proportions is relatively straightforward because homozygosity will be attained independently at the two loci. When the two loci are on the same chromosome, predicting offspring proportions is more complicated. Now there are not two but seven distinct genotypic configurations following prophase of meiosis I, with transitions between these states induced by crossovers between the two loci and between the loci and their centromere (Supplemental Material, Figure S1 in File S2). In Section 1 in File S1, the expected proportions of these configurations are derived both for fixed numbers of crossovers and assuming again a Poisson distribution of crossovers. Table 1 lists the corresponding proportions of genotypic configurations in a dually heterozygous mother for a number of scenarios of either complete linkage or absence of linkage. Also shown in Table 1 are the proportions of offspring genotypes resulting from each genotypic configuration under central fusion, terminal fusion, and gamete duplication automixis. Finally, averaging over all genotypic configurations yields the total expected offspring distribution under the different automixis and linkage scenarios, as shown again for extreme linkage scenarios and the three modes of automixis in Table 2.
A general result from this analysis is that, as expected, the total fraction of offspring that become homozygous at each locus is the same as predicted by the single-locus equations. Thus, in the case of central fusion automixis and denoting by A the locus that is more closely linked to the centromere than the other locus B, we have
(4)where 
is the expected number of crossovers between the centromere and locus A and 
the expected number of crossovers between loci A and B. However, the two rates at which homozygosity is attained are not independent. Instead, since the two loci are linked, the fraction of offspring that are homozygous at both loci is greater than what is expected for each locus individually:
(5)With terminal fusion, we obtain
(6)Neutral genetic variation
Let us assume an unstructured, finite population of females reproducing through automictic parthenogenesis. We consider a single locus at which new genetic variants that are selectively neutral can arise by mutation at a rate 
and we assume that each mutation produces a new allele (the “infinite alleles-model”). Heterozygosity in individuals may be lost through automixis (with probability γ), and genetic variation can be lost through drift (determined by population size N). We are interested in the equilibrium level of genetic variation that is expected in such a population.
A first quantity of interest is the heterozygosity 
i.e., the probability that the two alleles in a randomly chosen female are different. Following the standard approach for these type of models (e.g., Hartl and Clark 1997), the change in 
from one generation to the next can be expressed through the change in homozygosity, 
as
(7)Solving 
yields the equilibrium heterozygosity
(8)Here, the approximation ignores terms in 
and hence is valid for small mutation rates. This approximation illustrates that a balance is reached between the generation of new heterozygotes by mutation (at rate 2μ) and their conversion into homozygotes by automixis (at rate γ).
Second, we can calculate the probability 
that two alleles drawn randomly from two different females are different (the “population-level heterozygosity”). The recursion equation for 
is given by
(9)After substituting 
from Equation 8 for HI, the equilibrium is found to be
(10)Again, the approximation is valid for small 
and also assumes that N is large. Combined, these two quantities allow us to calculate 
the relative difference between population-level and individual-level heterozygosity at equilibrium:
(11)Note that as the mutation rate tends to zero, 
converges to 
and that 
is negative when 
Finally, we can calculate the probability 
that two randomly drawn individuals have a different genotype at the locus under consideration. Unfortunately, this seems to require a more elaborate approach than the simple recursion equations used to calculate 
and 
this approach is detailed in Section 2 in File S1. The resulting formula is rather cumbersome and not given here, but we can approximate 
assuming that the population size is not too small 
but not too large relative to the mutation rate. This yields the (still unwieldy)
(12)Figure 2 shows how the four statistics quantifying different aspects of genetic diversity depend on the rate 
at which heterozygosity in individuals is eroded, and compares them to the corresponding statistics in outbreeding sexual populations. Also shown in Figure 2 are diversity estimates from simulations (see Section 3 in File S1 for details), indicating that the analytical predictions are very accurate. It can be seen that both the within-individual and population-level heterozygosities decline with increasing 
Also, the former is greater than the latter for small values of 
resulting in negative values of 
but this pattern reverses for larger 
In Figure S2 in File S2, the same relationships are shown but expressed in terms of map distance for the case of central fusion automixis (see Equation 3). These plots illustrate that for the parameters assumed, large between-locus variation in equilibrium genetic diversity is only expected in the close vicinity of the centromeres. Finally, note that the equilibria shown in Figure 2 and Figure S2 in File S2 are reached much more slowly with low than with high values of 
(Figure S3 in File S2).
Equilibrium genetic diversity in neutrally evolving automictic populations, measured as (A) within-individual heterozygosity 
(B) population-level heterozygosity 
(C) relative reduction in heterozygosity 
and (D) diploid genotype-level diversity 
The solid lines in each plot show the analytical predictions for these statistics for varying rates 
at which heterozygosity is lost in the population, and for three mutation rates: 10−6 (blue, bottom), 10−5 (red, middle), and 10−4 (green, top). ●’s show the corresponding estimates from the simulations, and dashed lines show the corresponding expected values in outbreeding, sexual populations. Throughout, the population size was fixed to 
Note that most of the range of values for γ shown are relevant only for central fusion automixis 
while predictions for terminal fusion and gamete duplications are restricted to the rightmost part of the plots 
and 
respectively).
To gain further insight into the formulas derived above, it may be helpful to consider a few special cases.
Special case 1: γ = 0:
This case corresponds to strict clonal reproduction and has been studied previously (Balloux et al. 2003). In line with these previous results, all individuals are expected to eventually become heterozygous 
representing an extreme case of the Meselson effect (Welch and Meselson 2000; Normark et al. 2003). Furthermore, for the equilibrium population-level heterozygosity we get
(13)which is always greater than the corresponding genetic diversity in outbreeding sexual populations 
see also Figure 2B]. When 
is small, 
simplifies to 
(14)This is always negative because all individuals are heterozygous, but alleles sampled from different individuals may still be identical. Finally,
(15)i.e., 
is identical to the equilibrium heterozygosity in sexual populations. This is because with strict clonal reproduction, each mutation gives rise to not only a new allele but also to a new genotype. New diploid genotypes arise twice as often as new alleles in sexual populations, but this is exactly offset by twice the number of gene copies in sexual populations as genotypes in clonal populations.
Special case 2: γ = 1 (gamete duplication):
This represents the opposite extreme: barring new mutations, all heterozygosity is immediately lost. Thus, 
and, as a direct consequence, 
Moreover,
(16)Not surprisingly, 
because when all individuals are homozygous, comparing sampled alleles from different females and comparing genotype samples is equivalent. 
and 
are always lower than the expected heterozygosity in outbreeding sexual populations. Also note that with 
essentially the same approximations are obtained. This case is genetically equivalent to sexual reproduction with strict selfing, and indeed from the expressions in Balloux et al. (2003), the above expressions can also be derived under this assumption.
Special case 3: N → ∞:
As the equilibrium heterozygosity is not affected by population size, Equation 8 remains valid for 
As expected, both 
and 
converge to one as N goes to infinity. Finally, we have
(17)which is equal to the equilibrium individual-level homozygosity, 
and close to one for 
Effective population size
The results in the previous section can be put into perspective by considering the effective population size, Ne, a commonly used measure for the amount of genetic drift operating within a population. Several definitions for Ne have been proposed that may or may not be equivalent in different circumstances (Kimura and Crow 1963), but here the most natural choice is the coalescent effective population size (Nordborg and Krone 2002; Balloux et al. 2003; Sjödin et al. 2005). According to this, Ne is defined as 
where 
is the expected time until coalescence of two randomly chosen alleles in a population. In other words, given that 
in the standard coalescent with N diploid individuals, Ne is defined through a rescaling of time in a complex population model so that this population behaves like a Fisher–Wright population with respect to coalescent times.
For asexual populations, it is useful to further distinguish between a genotypic and an allelic effective population size (Balloux et al. 2003). The genotypic 
is based on coalescence of diploid genotypes, and in asexual populations is given simply by 
for both clonal (Balloux et al. 2003) and automictic populations. This is because instead of sampling 2N alleles every generation in sexual populations, N genotypes are sampled in clonal populations so that drift is stronger and coalescence times are shorter. This provides an alternative explanation for the expressions for 
obtained above. With clonal reproduction, the mutation rate for diploid genotypes is twice the mutation rate for alleles in sexual populations, but the effective population size is only half that of a sexual population, so that 
is the same as the equilibrium heterozygosity in outbreeding sexual populations. With gamete duplication, the genotypic mutation rate is effectively the same as the allelic mutation rate (because half of the mutations are immediately lost), so that 
can be obtained from the classic equilibrium heterozygosity in sexual populations by replacing N with 
In general, automixis itself is not part of the coalescence process from a genotypic point of view but rather contributes, along with mutation, to switches between different genotypes.
To obtain the allelic effective population size we can distinguish between two situations. First, the two alleles sampled may both reside within the same female. This occurs with probability 
and given that going backward in time the two alleles coalesce with probability 
in each generation, the expected coalescence time in this case is 
Second, the two alleles may be sampled from different females (with probability 
In this case, coalescence of the two genotypic lineages has to occur first (this will take on average N generations), and when this has happened the two alleles may either have also coalesced simultaneously (with probability 1/2), or it may again take an expected number of 
generations for coalescence to occur. In total we get
(18)which translates into an allelic effective population size of
(19)Thus, in contrast to the genotypic 
the allelic 
depends on 
so that different loci within the genome may experience different levels of genetic drift in relation to their distance from the centromere. With gamete duplication 
the allelic effective size is around 
i.e., the same as the genotypic effective size and half the effective size of a standard Fisher–Wright population. As 
decreases, the allelic effective population size increases, reaching that of the Fisher–Wright population for 
(which may be attained for some loci under central fusion automixis) and then increasing further to infinity as 
approaches zero (clonal reproduction; see also Balloux et al. 2003). This reflects the expectation that for clonally or near clonally inherited loci, the two gene copies within a genotypic line will never, or only after a very long time, coalesce. As shown in the previous section (and Figure 2, A and B), this results in within-individual heterozygosity levels near one and also high population-level heterozygosity.
Mutation–selection balance
To investigate the balance between the creation of deleterious alleles through mutation and their purging by natural selection in automictic populations, let us assume an infinitely large population and a single locus with two alleles a (wild type) and A (deleterious mutation). Wild-type aa individuals have a fitness of 
relative to heterozygotes with 
and mutant homozygotes with 
To keep the model tractable, I assume that the mutation rate 
is small so that at most one mutation event occurs during reproduction, and that there is no back mutation from mutant to wild-type allele. Automixis operates as in the previous section, with heterozygotes producing a fraction 
of either homozygote. Assuming a life-history order of selection, mutation, and automixis, the recursion equations for the frequencies of the two mutant genotypes can be expressed as
(20)Solving 
yields the frequencies of the mutant genotypes under the mutation–selection equilibrium. Unfortunately, the resulting formulas are rather uninformative and not given here. However, we can make progress by linearizing Equations 20 under the assumption that 
and 
are all ≪1 and of the same order. These assumptions will be met if 
and if 
is not too small. This approximation reads
(21)and has the solution
(22)It may be informative to consider a few special cases where tractable results can also be obtained directly from the solution to system (20).
Special case 1: Clonal reproduction (γ = 0):
In this case, the equilibrium that will be attained depends on the magnitude of the mutation rate. First, when the mutation rate is small, 
we get
(23)At this equilibrium, the population will consist mostly of mutation-free aa individuals, with some heterozygotes and very few AA homozygotes. Note that 
is approximately the same as the equilibrium frequency of heterozygotes in an outbreeding sexual population.
Second, when 
the equilibrium is given by
(24)No mutation-free aa individuals persist in the population in this case (since 
Intuitively, this situation arises when selection against heterozygotes is so weak relative to the mutation rate that eventually all aa individuals are converted into heterozygotes and a mutation–selection balance is attained between Aa and AA individuals. This balance is then analogous to the standard mutation–selection balance in haploid populations. Indeed, after renormalizing all fitness values with the fitness of heterozygotes and defining an adjusted selection coefficient against AA homozygotes, 
the equilibrium frequency of homozygous mutants can be expressed as 
Finally, when 
the mutation pressure outweighs selection completely and the mutant homozygotes will become fixed in the population 
Special case 2: Gamete duplication (γ = 1):
At the opposite extreme, when all heterozygotes are immediately converted into homozygotes, the equilibrium frequencies are given by
(25)It is clear that in this case, selection against heterozygotes and thus the dominance coefficient h is irrelevant. Mutation-free aa homozygotes produce heterozygote mutant offspring at a rate 
per generation, but these heterozygotes are immediately converted into aa and AA offspring, each with probability 1/2. Thus, the effective rate at which AA offspring are produced is 
and the attained mutation–selection balance is identical to the one attained in haploid populations.
Special case 3: Recessive deleterious mutations (h = 0):
With arbitrary values of 
but strictly recessive deleterious mutations, the equilibrium is given by
(26)Thus, the equilibrium frequency 
of mutant homozygotes is identical to the one expected in sexual populations with recessive deleterious mutations (corresponding to an equilibrium allele frequency of 
The equilibrium frequency of heterozygotes is a decreasing function of 
and can be either higher or lower than the corresponding equilibrium frequency of heterozygotes in sexual populations.
Figure 3 shows equilibrium frequencies of the Aa and AA genotypes under mutation–selection balance for recessive, partially recessive, semidominant, and dominant mutations, and compares these frequencies to the corresponding frequencies in sexual populations. For 
the frequency of heterozygotes is generally lower than in sexual populations, whereas the frequency of AA homozygotes is higher than in sexual populations. Related to this, the equilibrium frequencies in automictic populations are generally much less sensitive to the dominance coefficient h than in sexual populations because 
implies that selection against heterozygotes is much less important than in sexual populations. Also note that populations reproducing by strict selfing (corresponding to 
are characterized by very similar frequencies of AA homozygotes as automictic populations with high values of 
while the frequencies of heterozygotes are very low in both cases.
Equilibrium genotype frequencies under mutation–selection balance with different homozygosity acquisition rates γ, and for mutations that are (A) recessive 
(B) partially recessive 
(C) semidominant 
and (D) dominant 
In each plot, the bold solid line gives the equilibrium frequency of Aa heterozygotes and the bold dashed line the equilibrium frequency of AA homozygotes. For comparison, the thin lines show the corresponding frequencies in outbreeding sexual populations. [Note that the dashed thin line is on top of the dashed bold line in (A) and outside the plotting range in (D).] Other parameters take the values 
and 
The equilibrium frequencies can also be used to calculate the mutational load 
i.e., the relative reduction in the mean fitness of the population caused by recurrent mutation. This quantity can be expressed as
From this and Equations 25 and 26 it can be deduced that both with gamete duplication 
and recessive mutations 
the genetic load in the population is given by the mutation rate 
the same as for recessive mutations in sexual diploid populations and also the same as in haploid populations. In general, 
will be greater than 
but always lower than 
and, interestingly, often lower than the genetic load in sexual populations (Figure S4 in File S2). This is in contrast to previous theoretical studies on the mutational load in clonal diploids in which the maintenance of heterozygosity caused the asexual populations to accumulate a higher load than the sexual populations (Chasnov 2000; Haag and Roze 2007). It is important to note, however, that the simple results derived here do not account for finite population size and interference between multiple loci, which may have a strong impact on the mutation–selection balance and the mutational load (Glémin 2003; Haag and Roze 2007; Roze 2015).
Overdominance
When there is overdominance (i.e., a heterozygote fitness advantage over the homozygote genotypes), it is useful to parameterize the fitness values as 
and 
with 
The recursion equation for the genotype frequencies 
and 
can then be expressed as
(27)Solving 
yields the following polymorphic equilibrium:
(28)This equilibrium takes positive values for 
and 
and stability analysis indicates that this is also the condition for the equilibrium to be stable. [The eigenvalues of the associated Jacobian matrix are 
and 
Thus, overdominant selection can maintain heterozygotes in the face of erosion by automixis if the selection coefficient against both homozygotes is greater than the rate at which heterozygosity is lost. This result has previously been conjectured by Goudie et al. (2012) on the basis of numerical results of a similar model, and more complex expressions for equilibrium (28) have been derived by Asher (1970).
Depending on 
the equilibrium frequency of heterozygotes can take any value between 0 
and 1 (when 
i.e., with clonal reproduction). This is shown in Figure 4 and contrasted with the equilibrium frequency in outbreeding sexual populations, given by 
Equilibrium heterozygote frequencies under overdominant selection and for given rates γ at which heterozygotes are converted into homozygotes. The bold lines show these frequencies under automictic reproduction and for 
(solid line), 
(dashed line), and 
(dotted line). For comparison, the thin lines show the corresponding frequencies in outbreeding sexual populations.
We can also calculate how much the mean fitness in the population at equilibrium is reduced by automixis compared to clonal reproduction. Provided that 
this “automixis load” is given by
(29)This simple formula parallels the classic result that the mutational load in haploid populations is given by the mutation rate and thus shows that automixis acts like mutation in producing two genotypes (AA and aa), of inferior fitness from the fittest genotype (Aa), that are then purged by natural selection. The genetic load can be either smaller or greater than the corresponding segregation load in a sexual population. More precisely, when 
(Crow and Kimura 1970), there will be more heterozygotes in the automictic population and their genetic load will be lower than in the sexual population, and vice versa. When 
the heterozygotes are lost from the population and either the aa (if saa < sAA) or the AA genotype (if saa > sAA) will become fixed. In this case, we obtain the largest possible load, 
.
Associative overdominance
In addition to overdominance, heterozygosity could also be maintained through off-phase recessive deleterious mutations at tightly linked loci (Frydenberg 1963; Ohta 1971), and this has been proposed to explain heterozygosity in the Cape honeybee (Goudie et al. 2014). Consider a recently arisen lineage reproducing through central fusion automixis in which, by chance, the founding female carries a strongly deleterious recessive mutation A on one chromosome and another strongly deleterious recessive mutation B at a tightly linked locus on the homologous chromosome. Thus, the genetic constitution of this female is AabB. Then, the vast majority of offspring that have become homozygous for the high-fitness allele a are also homozygous for the deleterious allele B and vice versa, so that linkage produces strong indirect selection against both aa and bb homozygotes.
To investigate this mechanism, numerical explorations of a two-locus model of an infinitely large population undergoing selection and reproduction through automixis were performed. This model builds upon the expressions for loss of heterozygosity in the presence of recombination between a centromere and two loci derived above (for details see Section 4 in File S1). Heterozygosity at either locus entails a reduction of fitness of 
whereas homozygosity for the deleterious alleles reduces fitness by 
in each locus. Fitness effects at the two loci are multiplicative (i.e., no epistasis). An example run is shown in Figure 5A. As can be seen, the AabB genotype is maintained at a high frequency for a considerable number of generations (in automixis–selection balance with the two homozygous genotypes AAbb and aaBB) before it is eroded by recombination between the two loci and the aabb genotype spreads to fixation. We can also treat the two linked loci as a single locus and use the equilibrium frequency of heterozygotes derived above for single-locus overdominance to estimate the quasi-stable frequency of the AabB genotype before it is dissolved. Specifically, this frequency can be approximated by Equation 28 following substitution of 
for 
for 
and 
for the expression in Equation 3, yielding
(30)For simplicity, this approximation assumes complete recessivity 
but partial recessivity could also readily be incorporated. As shown in Figure 5A, this approximation is very close to the quasi-stable frequency of the AabB genotype obtained numerically.
Maintenance of strongly deleterious mutations (A and B) in an AabB genotype through associative overdominance with central fusion automixis. (A) Example evolutionary dynamics where solid lines show the four predominant genotypes in the population and the dashed blue line gives the approximation for the quasi-stable frequency of the AabB genotype. Parameters take the values 
and 
(B) Time until dissolution of the AabB genotype for different mean crossover numbers 
(between centromere and locus A) and 
(between loci A and B). Values shown are the number of generations until the frequency of the AabB genotype has dropped from 1 to 0.01 in the population. Other parameters take the values 
and 
Next, we can ask for how long the AabB genotype is expected to persist in the population. To address this question, screens of the parameter space with respect to the two mean crossover numbers were performed. The recursion equations were initiated with only AabB individuals present in the population and iterated until their frequency dropped below 0.01. The number of generations this took for different mean numbers of crossovers 
between locus A and the centromere and different numbers of crossovers 
between loci A and B are shown in Figure 5B. It can be seen that, provided the two loci are both tightly linked to the centromere, central fusion automixis can indeed maintain the polymorphism for many generations. The same principle also applies to terminal fusion automixis and sexual reproduction, but here the deleterious recessive mutations are only maintained for very short time periods (results not shown).
Spread of beneficial mutations
To better understand adaptive evolution in automictic populations, consider first a deterministic single-locus model without mutation and with relative fitness 
and 
for heterozygotes and AA homozygotes, respectively. Analogously to Equations 20 for deleterious mutations (but assuming a mutation rate of zero), the recursion equations for this model can then be written as
(31)Assuming that both the Aa and the AA genotype are rare in the population and that s is small, these recursion equations can be approximated by
(32)This linear system of recursion equations can be solved, and if we further assume that initially there are only one or few heterozygote mutants but no AA homozygotes 
this solution becomes
(33)with 
These expressions demonstrate that when 
is large relative to the selection benefit of heterozygotes—more precisely when 
or 
—the heterozygotes will not be maintained and the beneficial mutation will instead spread as a homozygous genotype through the population. Thus, in this case we have for sufficiently large t:
(34)Here, h and 
determine how efficiently the initial heterozygotes are maintained and converted to homozygotes, but only s determines the actual rate at which the beneficial mutation spreads. By comparison, a beneficial mutation in an outbreeding sexual population will initially be found in heterozygotes only, with
(35)Thus, the rate at which the mutation spreads in sexual populations is determined by the fitness advantage in heterozygotes only, which means the mutation will always spread at a lower rate than in automictic populations. Nevertheless, the heterozygotes in sexual populations have a “head start” relative to the homozygotes in automictic populations (see fraction in Equation 34), which results from the fact that only half of the original heterozygotes are converted into homozygotes. This means that with high dominance levels h, it might still take some time until a beneficial allele reaches a higher frequency in an automictic compared to a sexual population.
When 
both heterozygotes and AA homozygotes will spread simultaneously in the population and, for very small 
it may take a long time until the homozygotes reach a higher frequency than the heterozygotes. In the extreme case of 
(clonal reproduction), heterozygote frequency increases by a factor of 
in each generation (i.e., at the same rate as in outbreeding sexuals), and no homozygotes are produced.
The above considerations apply only to the early phase of the deterministic spread of a beneficial mutation destined to spread in a population. In a more realistic setting involving random genetic drift, we can also ask what the fixation probability of beneficial mutations is in automictic populations. Assuming that offspring numbers of rare mutant females are distributed independently, we can treat the spread of beneficial mutations as a multi-type branching process (e.g., Allen 2003, Chap. 4). (Note that this approach is valid for large populations; for small populations different approaches that take the effective population size into account will provide better approximations.) Building on Equation 32, the mean number 
of offspring of type j (1 = Aa, 2 = AA) produced by a mother of type i is given by the matrix
(36)Assuming that offspring numbers follow Poisson distributions with these means, we arrive at the following probability-generating functions for the branching process:
(37)Simultaneously solving
(38)then yields the fixation probabilities 
and 
when there is initially a single Aa or AA mutant individual, respectively. Equations 38 can only be solved numerically. However, assuming weak selection, series expansion of 
and retaining only quadratic and lower terms gives the approximation 
mirroring a well-known result in haploid populations. Substituting 
for 
in the expression for 
expanding, and again retaining only quadratic or lower terms then gives the biologically more interesting
(39)with placeholder 
When 
is not too small, this expression can be further approximated by Taylor expansion in 
(discarding terms above second order) to yield
(40)Figure 6 illustrates both Equation 39 and numerical solutions of Equations 38 for the fixation probability of a new beneficial mutation. Figure 6 also shows that unless selection is strong, there is a good correspondence between these results based on the branching process approximation and simulation results. It can be seen that for recessive and partially recessive mutations, the fixation probability increases with increasing 
and is greater than the fixation probability in outbreeding sexual populations (Figure 6, A and B). Beneficial mutations with additive effects are relatively insensitive to 
(Figure 6C), and for dominant mutations the fixation probability decreases with increasing 
The approximation in Equation 40 also illustrates the dependency of 
on 
for different dominance levels. In all cases, the fixation probability converges to 
as 
approaches 1. This is rather intuitive as with 
(i.e., gamete duplication), an Aa female produces an AA offspring individual with a probability of approximately 1/2 in the first generation, upon which the AA genotype then fixes in the population with probability 
When 
corresponding to strict selfing, the fixation probability is already close to 
as has been derived previously (Charlesworth 1992).
Fixation probabilities of beneficial mutations arising as a single heterozygous genotype. Results are shown for varying rates 
of loss of heterozygosity and when mutations are (A) recessive (h = 0), (B) partially recessive (h = 0.05), (C) additive (h = 0.5), or (D) dominant (h = 1). Solid lines show the numerical solutions of Equation 36 and dotted lines show the approximation in Equation 37, with s = 0.01 (blue, bottom), s = 0.05 (red, middle), or s = 0.1 (green, top). Dashed horizontal lines show the corresponding fixation probabilities in outbreeding sexual populations, approximated as 2hs (Haldane 1927). ●’s show simulation results obtained by iterating Equation 29 with a multinomial sampling step inserted in each generation and assuming a population size of N = 10,000; these simulations were run 50,000 times for each parameter combination until the beneficial allele became extinct or fixed.
The above considerations demonstrate that unless beneficial mutations are completely dominant, their deterministic spread is expected to be faster in automictic than in either clonal or sexual populations. For recessive or partially recessive beneficial mutations, this also translates into a higher probability of fixation. To what extent do these results hold when more than one locus is considered? It is well known that asexually reproducing populations may fix beneficial mutations more slowly because of clonal interference, i.e., competition between simultaneously spreading beneficial mutations that, in the absence of recombination, cannot be brought together into the same genome (Fisher 1930; Muller 1932). Despite the term “clonal interference,” this mechanism should also operate in automictic populations, and we can ask whether and under what conditions the decelerating effect of clonal interference on the speed of adaptive evolution can offset the beneficial effects of turning heterozygotes with one beneficial mutation into homozygotes with two beneficial mutations.
To address this question, numerical investigations of a two-locus model involving recombination, automixis, selection, and drift were performed. (Note that clonal interference only operates in finite populations subject to random genetic drift and/or random mutations.) Full details of this model can be found in Section 5 in File S1. The results of a screen of the two crossover numbers 
(between locus A and the centromere) and crossovers 
(between loci A and B) are shown in Figure S5 in File S2. It appears that unless 
is very small, recessive beneficial mutations spread considerably faster in automictic than in sexual populations, despite clonal interference in the former. With additive fitness effects, the difference between automictic and sexual populations is less pronounced and the beneficial effects of recombination become apparent when 
is very small. Perhaps surprisingly, the mean number 
of crossovers between the two loci under selection, which determines the recombination rate in the sexual population, plays only a minor role and needs to take low values for recessive beneficial mutations to spread slightly faster in sexual than automictic populations (bottom-left corner in Figure 6A). This is because although increasing values of 
lead to faster spread of the beneficial mutations in sexual populations, this effect is only weak compared to the accelerating effect of increasing 
in automictic populations.
The results obtained in this section regarding the spread of beneficial mutations are only a step toward understanding adaptive evolution in automictic populations. Further progress should be made by deriving expressions for fixation and extinction times (see, e.g., Mandegar and Otto 2007 and Glémin 2012 for the related cases of clonal reproduction with mitotic recombination and selfing, respectively), by considering many loci (see Pamilo et al. 1987 for some results on automictic populations for mutations with additive fitness effects), and by developing models for the recurrent spread of beneficial mutations with variable fitness effects (see, e.g., Kim and Orr 2005 for a comparison between sexual and clonal populations).
Selection on crossover rates
We finally turn to the question of whether natural selection is expected to reduce crossover rates in automictic populations. Let us first consider a population reproducing by central fusion automixis in which heterozygosity at a given locus is maintained through overdominant selection. Combining the results from Equations 3 and 29, the mean fitness of a resident population with a mean crossover number 
is given by
(41)Since reproduction is asexual and assuming a dominant crossover modifier allele, the selection coefficient 
for a mutant genotype with a different crossover rate can be obtained simply by comparing the mean fitness of the resident and the mutant lineage. [This is in contrast to recombination rate evolution in sexual populations, where a much more sophisticated approach is required (Barton 1995).] If the factor by which crossover numbers are altered is denoted by 
(i.e., the mutant mean number of crossovers is 
this yields
(42)where the approximation is valid for small 
As expected, any mutant in which crossover rates are suppressed 
is selectively favored 
In the extreme case of 
(complete crossover suppression),
(43)which ranges from around 
when the initial crossover rate is already very small, to 
when 
is very large.
With terminal fusion automixis, we obtain
(44)Again, the approximation is valid for small 
(but note that conditions where overdominance stably maintains heterozygosity are rather limited in this case; see Equation 26). Increases in crossover rates are selected for with terminal fusion, but even when there are many crossovers between the focal locus and its centromere, a substantial genetic load 
will persist.
We can also ask how selection should operate on crossover rates in automictic populations evolving under mutation–selection balance. Without answering this question in any quantitative detail, we can note that since the equilibrium genetic load decreases with increasing 
(Figure S4 in File S2), there should be selection for increased crossover rates in populations with central fusion automixis and selection for decreased recombination rates in populations with terminal fusion automixis. However, given that genetic load is always in the range between 
and 
selection for increased crossovers will be only very weak on a per-locus basis 
Stronger selection is expected when there are preexisting deleterious mutations at high frequencies (e.g., following the emergence of a new, genetically uniform automictic lineage). For example, with partially recessive mutations and central fusion automixis, there may initially be strong selection against crossovers (avoiding the unmasking of the mutation in heterozygotes), followed by selection for increased crossover rates (enabling more efficient purging of the mutation). Preliminary numerical investigations competing two lineages with different crossover rates confirm these predictions on selection on crossover rates (e.g., Figure S6 in File S2). However, it will be important to study this problem more thoroughly in a multi-locus model and with finite populations so that, for example, the impact of stochastically arising associative overdominance can also be ascertained.
Data availability
The authors state that all data necessary for confirming the conclusions presented in the article are represented fully within the article. Most results were obtained using Mathematica v.10 (Wolfram Research Inc.); the Mathematica notebook is available at https://github.com/JanEngelstaedter/AutomixisEvolution.
Discussion
Automixis as a viable system of reproduction
Automixis is a peculiar mode of reproduction. Not only are, as with other modes of asexual reproduction, the benefits of recombination forfeited, but the fusion of meiotic products to restore diploidy also means that heterozygosity can be lost at a high rate. This raises intriguing questions as to why automixis has evolved numerous times and how stably automictically reproducing populations can persist. Previous work has shown that the loss of complementation faced by a newly evolved automictically reproducing female can cause severe reductions in fitness that may exceed the twofold cost of sex, and are likely to severely constrain the rate at which sex is abandoned (Archetti 2004, 2010; Engelstädter 2008). However, the results obtained here indicate that once an automictic population is established, it may persist and in some respects even be superior to clonal or sexual populations. In particular, neutral genetic diversity will be lower in automictic than in clonal populations but may still be greater than in sexual populations, the mutational load will generally be lower in automictic than in both sexual and clonal populations (unless mutations are completely recessive), and the genetic load caused by overdominant selection can be lower in automictic than in sexual populations.
Empirical examples confirming that automicts can be highly successful at least on short to intermediate timescales include the Cape honeybee clone (which has been spreading for >25 years) (Goudie and Oldroyd 2014), the invasive ant Cerapachys biroi (which has been reproducing asexually for at least 200 generations) (Wetterer et al. 2012; Oxley et al. 2014), and Muscidifurax uniraptor wasps which have been infected by parthenogenesis-inducing Wolbachia for long enough that male functions have degenerated (Gottlieb and Zchori-Fein 2001). Of course, automictic populations still suffer from the lack of recombination and hence long-term consequences such as the accumulation of deleterious mutations through Muller’s ratchet or reduced rates of adaptation because of clonal interference. It therefore does not come as a surprise that, like other asexuals, automictic species tend to be phylogenetically isolated (Schwander and Crespi 2009). One exception to this rule are the oribatid mites, in which ∼10% out of >10,000 species reproduce by automixis and radiations of automictic species have occurred (Domes et al. 2007; Heethoff et al. 2009). To better understand the long-term dynamics of adaptation and mutation accumulation in automictic populations, it would be useful to develop more sophisticated models than presented here that incorporate multiple loci and random genetic drift.
Relationship to other genetic systems
There is a bewildering diversity of genetic systems that have similarities to automixis. To discuss the relationship of the results obtained here with prior work it may be useful to group these genetic systems into two classes. The first are systems that are mechanistically distinct from automixis but are genetically equivalent. This includes systems in animals and plants where there is no fusion of meiotic products but some other meiotic modification that has the same consequences, and also systems of intratetrad mating. The results obtained here are thus directly applicable, and previous theoretical work on such systems can directly be compared to the work presented here. For example, parthenogenesis in Daphnia pulex has been reported to proceed through a modified meiosis in which the first anaphase is aborted halfway, homologous chromosomes are rejoined, and the second meiotic division proceeds normally (Hiruta et al. 2010). Similarly, some forms of apomixis in plants (meiotic diplospory) are also achieved by suppression of the first meiotic division (Gustaffson 1931; van Dijk 2009). These modifications of meiosis are genetically equivalent to central fusion automixis and can, through complete suppression of recombination, also lead to clonal reproduction. Intratetrad mating is commonly found in many fungi, algae, and other organisms and is achieved through a variety of mechanisms (Kerrigan et al. 1993; Hood and Antonovics 2004). Provided the mating-type locus is completely linked to the centromere, intratetrad mating is genetically equivalent to central fusion automixis (Antonovics and Abrams 2004). If the mating-type locus is not closely linked to the centromere, the outcome would still be equivalent to automixis but with a mixture of terminal and central fusion, depending on whether or not there has been a recombination event between the mating-type locus and the centromere. Finally, systems of clonal reproduction with mitotic recombination should also be equivalent to central fusion automixis; here, heterozygosity is also lost with probability 1/2 during one of the mitoses in the germline when there was a recombination event between a locus and its associated centromere (Mandegar and Otto 2007).
The second class of systems comprises those that are very similar to automixis but equivalent only when a single locus is considered. This means that many of the results reported here (e.g., on neutral variation, mutation–selection balance, and overdominance) can still be applied. For example, clonal populations in which there is occasional, symmetrical gene conversion can be considered genetically equivalent to the single-locus models considered here, with the rate of loss of heterozygosity 
replaced by the gene conversion rate. Gene conversion has been reported in several parthenogenetic animals (Crease and Lynch 1991; Schon and Martens 2003; Flot et al. 2013), and recently a number of results concerning coalescent times and patterns have been derived for such systems (Hartfield et al. 2016).
Populations that reproduce exclusively by selfing also belong into this class; such populations are characterized by a rate of heterozygosity loss of 
Thus, some loci in populations reproducing by terminal fusion automixis are expected to evolve in the same way as selfing populations, and loci with high values of 
(including all loci in populations reproducing by gamete duplication) may exhibit levels of genetic variation that are, in practical terms, indistinguishable from those under pure selfing. It should be emphasized, however, that selfing in general is distinct from automixis: in essence, the difference is that alleles are sampled either with (selfing) or without (automixis) replacement from the meiotic products. As a consequence, the genome-wide distribution of heterozygosity levels should be characteristically dependent on genetic distance from the centromeres in automictic but not in selfing populations, and this can be exploited to distinguish between these two systems (Svendsen et al. 2015). The only exceptions to this rule are automictic populations reproducing by gamete duplication (where 
for all loci), and when gametic products fuse randomly (resulting in 
for all loci).
Populations with mixed automictic and sexual reproduction
The models presented here assume populations that reproduce exclusively through automixis. Although several species with exclusively automictic reproduction have been reported, many other species exhibit different forms of mixed sexual and automictic reproduction. The simplest case is one where a lineage of automicts competes with sexual conspecifics but where there is no gene flow between these two populations. Such a situation is found in the Cape honeybee, A. mellifera capensis, in which a subpopulation (the clone) reproduces through central fusion automixis and parasitizes colonies of a sexual subspecies, A. mellifera scutellata (Goudie and Oldroyd 2014). In principle, the results presented here could be used to predict the outcome of such competitions by comparing population mean fitness of sexual and automictic populations. However, the case of the Cape honeybee is fraught with a number of additional complexities, including both honeybee characteristics such as eusociality and the complementary sex determination system, as well as the parasitic nature and epidemiological dynamics of the clone. This will make it necessary to develop specifically tailored models that incorporate both the evolutionary genetics consequences of automixis explored in the present article and the ecological and genetic idiosyncrasies of the clone (Martin et al. 2002; Moritz 2002).
More complicated is the case of gene flow between the sexual and automictic subpopulations. This can occur, for example, when otherwise automictic females occasionally produce males. Provided these males are viable and fertile, they may mate and produce offspring with the sexual females. This will not only introduce genetic material from the asexual into the sexual populations, but it may also lead to the emergence of new automictic lineages because the males may transmit the genes coding for automictic reproduction to their female offspring. Such cases of “contagious parthenogenesis” (Simon et al. 2003) associated with automixis have been reported in the parasitoid wasp Lysiphlebus fabarum (Sandrock and Vorburger 2011; Sandrock et al. 2011) and Artemia brine shrimps (Maccari et al. 2014). Some aspects of the evolutionary dynamics of such systems have been studied (Engelstädter et al. 2011), but population genetic processes such as the ones studied here remain to be investigated.
Finally, there are many species in which there are no clear sexual and asexual subpopulations but where females can reproduce both sexually and through automixis. This includes, for example, the majority of Drosophila species where parthenogenesis has been reported (Markow 2013), and also a number of vertebrates with facultative parthenogenesis (Lampert 2008). It is expected that sexual populations capable of occasional automictic reproduction should not differ much from sexual populations in terms of population genetic processes. One exception is that rare automixis may facilitate the colonization of previously uninhabited areas. Distinguishing the genomic signature of the resulting automictically arisen population bottlenecks from those of “conventional” bottlenecks will be challenging but may be feasible with data on genome-wide levels of heterozygosity (see also Svendsen et al. 2015). On the other hand, rare sex in predominantly automictic populations is expected to have a great impact as the mixing of lineages may efficiently counteract clonal interference and Muller’s ratchet (Hojsgaard and Horandl 2015).
Conclusions
In this study, a number of theoretical results regarding basic population genetic processes in automictic populations were derived for both neutral and selective processes. A general conclusion that emerges is that, in analogy to strong levels of inbreeding, automictic reproduction is difficult to evolve but once established may be viable on intermediate timescales and even has advantages compared to clonal and sexual reproduction. Future theoretical work is still necessary to elucidate long-term evolutionary patterns of automictic species, such as the rate of mutational meltdown under Muller’s ratchet or the dynamics of adaptation.
Acknowledgments
I thank Christoph R. Haag, Nicholas M. A. Smith, and two anonymous reviewers for their insightful comments on the manuscript. One of the anonymous reviewers provided particularly helpful advice on obtaining compact approximations for some of the analytical results. I thank Isabel Gordo and the Instituto Gulbenkian de Ciência (Portugal) for hosting me during the final stages of preparing this manuscript. The author declares no conflict of interest.
Footnotes
Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.196873/-/DC1.
Communicating editor: J. Hermisson
- Received October 17, 2016.
- Accepted March 21, 2017.
- Copyright © 2017 by the Genetics Society of America
