Abstract
Here I present a simple population genetic model to investigate the evolution of polymorphic haploiddiploid life cycles. The key feature of the model is the assumption of mutation occurring during meiosis. I show that, in addition to regions favoring haploid or diploid life cycles, there are substantial regions of the parameter space under which polymorphic haploiddiploid life cycles are expected to evolve.
ALL meiotic organisms spend some proportion of their life as a haploid and as a diploid. In diploid organisms, mitotic divisions are essentially restricted to the diploid phase. In haploid organisms, mitotic divisions are restricted to the haploid phase. Haploiddiploid life cycles are those in which mitotic divisions occur in both the haploid and diploid phases within a population (Bell 1994). Since the life cycle is one of the most fundamental attributes of an organism, understanding the variation seen among meiotic organisms in their life cycles is an important problem in evolutionary biology. In particular, we would like to identify and determine the relative importance of the factors that affect the evolution of the life cycle.
I distinguish two types of haploiddiploid life cycle. In biphasic species (Figure 1a), individuals have mitotic divisions in the diploid phase, undergo meiosis, and then have mitotic divisions in the haploid phase. In polymorphic species (Figure 1b), individuals undergo mitotic divisions in either the haploid or the diploid phase.
Thus, after syngamy individuals can either undergo meiosis immediately to produce haploid offspring who will undergo somatic development, or meiosis can be delayed such that somatic development occurs in the diploid phase. In this article, I seek to understand under what conditions polymorphic haploiddiploid life cycles are expected to evolve in response to meiotic mutation. While previous work has considered the evolution of polymorphic life cycles in response to mutation, none has specifically addressed meiotic mutation, instead implicitly focusing on mitotic mutation.
Previous models: The relative advantages of diploid vs. haploid life cycles have been considered in several studies. Some of these have used the relative fitness of a haploid vs. a diploid population (Charlesworth 1991; Kondrashov and Crow 1991) to understand conditions favoring one ploidy over the other and, as such, have not considered the evolution of a haploiddiploid life cycle. Other models have explicitly allowed a haploiddiploid life cycle to be a possible evolutionary outcome (Perrotet al. 1991; Bengtsson 1992; Goldstein 1992; Otto and Goldstein 1992; Bell 1994; Michod and Gayley 1994; Orr and Otto 1994; Otto and Marks 1996). Models addressing polymorphic life cycles have uniformly found that a haploiddiploid cycle is unable to evolve. Instead a population is expected to evolve to a haploid or a diploid life cycle depending on the values of the parameters. The few models addressing biphasic life cycles (Jenkins 1993; Jenkins and Kirkpatrick 1994; Hughes and Otto 1999) have found that a haploiddiploid cycle can evolve, at least under certain fitness functions.
With mutation occurring primarily during meiosis, I find that there is a significant region of the parameter space in which a polymorphic haploiddiploid life cycle can evolve. This is because meiotic mutation leads to a negative frequency dependence between the advantage of diploidy and the proportion of the population that is currently diploid.
THE MODEL
Apart from the timing of mutation, the model follows that of Perrot et al. (1991) and Otto and Goldstein (1992). I consider an organism with synchronized mating such that fusion of haploid gametes occurs at a particular time set by external signals such as day length. Following the fusion of haploid gametes to form diploid zygotes, a cell either immediately undergoes meiosis, or meiosis is delayed until just prior to the next episode of mating. Ploidy level is thus controlled by the timing of meiosis (Perrotet al. 1991; Otto and Goldstein 1992). Delaying meiosis results in an individual that enters adulthood and undergoes selection as a diploid. By undergoing meiosis immediately following zygote formation, individuals are produced that undergo selection as haploids (Figure 2).
The probability that a cell fails to undergo meiosis immediately following zygote formation and thus enters adulthood as a diploid is controlled by a modifier locus, C. The C_{1}C_{1}, C_{1}C_{2} and C_{2}C_{2} genotypes at this locus cause a cell to remain diploid with probability d_{11}, d_{12}, and d_{22}, respectively. The modifier heterozygote shows intermediate dominance (i.e., d_{11} < d_{12} < d_{22} or d_{11} > d_{12} > d_{22}), and differences between modifier genotypes, in terms of the probability of undergoing early meiosis, are assumed to be small such that terms of order (d_{ij} – d_{kl})^{2} can be ignored.
Fitness of adults is determined by a viability locus that segregates a favored allele, A_{0}, and a deleterious allele, A_{1}. Selection is such that the A_{0}A_{0} and A_{0} genotypes have fitness 1; A_{1}A_{1} and A_{1} have fitness 1 – s, and A_{0}A_{1} has fitness 1 – hs (0 ≤ h ≤ 1, 0 ≤ s ≤ 1). Mutation is assumed to occur during meiosis at rate μ from A_{0} to A_{1}. Unlike previous models, mitotic mutation is ignored. The recombination rate r between the modifier and viability loci can take any value (0 ≤ r ≤ ½). See Figure 2 for an overview of the model.
RESULTS
Setting x_{1}, x_{2}, x_{3}, and x_{4} as the frequencies of A_{0}C_{1}, A_{1}C_{1}, A_{0}C_{2}, and A_{1}C_{2}, respectively, just prior to syngamy, the recursions for the model simplify to
When a new allele is introduced at low frequency at the modifier locus into a population at the equilibrium given in Equations 1, recursions in the rare genotypes (x_{3} and x_{4}) can be linearized, since we can ignore terms that are squared in these frequencies, to give
The roots of the characteristic equation of these linear recursions give the eigenvalues. If the leading eigenvalue is >1, the introduced allele increases in frequency. If the introduced modifier did not alter ploidy levels, such that it was neutral, the leading eigenvalue would equal 1 and the other eigenvalue would be positive and <1 (from PerronFrobenius theorem; Gantmacher 1959). Under weak selection, the leading eigenvalue is close to 1 in value. If the leading eigenvalue is >1, then the sign of the characteristic equation evaluated at 1, C(1), is negative. If the leading eigenvalue is <1 then C(1) is positive. Thus, the sign of C(1) determines stability when selection is weak. Evaluating C(1) gives the condition for the invasion of an introduced rare modifier allele (C_{2}) as
A change of basis (Uyenoyama and Bengtsson 1989; Uyenoyama 1991) was performed such that the invasion criterion could be partitioned into a term involving average fitness and a term due to associations that arise between the two loci during invasion of the C_{2} allele (see appendix). The new basis is such that one axis represents the frequency of the C_{2} allele (designated the p axis) and the other axis represents the standard measure of linkage disequilibrium, D, between the two loci (see Crow and Kimura 1970). With an appropriate choice of vector {p*, D*}^{T} in the new basis and assuming weak selection (see appendix), this analysis gives the condition for the increase of C_{2} when rare as
The first term of inequality (3) measures the effect of changing the ploidy level on mean fitness and, as such, ignores associations between the two loci. An expression for (V_{D} – V_{H}) can be calculated as
DISCUSSION
The results from this model agree with previous work (Perrotet al. 1991; Bengtsson 1992; Otto and Goldstein 1992; Otto 1994; Otto and Marks 1996) in that recessive mutations (small h) and looser linkage (r close to ½) both favor diploid life cycles while dominant mutations (large h) and tight linkage (r close to 0) both favor haploid life cycles. However, in a departure from previous work, polymorphic haploiddiploid life cycles are expected to evolve for a substantial region of the parameter space. In particular, a combination of loose linkage, strong selection, and mutations that are not too recessive favors a polymorphic haploiddiploid life cycle (see Figure 4).
Mean fitness of haploids vs. diploids: In previous models that have examined the evolution of haploiddiploid polymorphic life cycles, the frequency of deleterious alleles in haploids and in diploids entering selection is the same. As such, the only difference between haploids and diploids is in how those deleterious alleles are subjected to selection. In haploids, the deleterious allele is selected against in the haploid genotype and as such suffers a fitness cost equal to s. In diploids, the deleterious allele occurs primarily in the heterozygote and as such suffers a fitness cost equal to hs. The frequency of the heterozygote in diploids is approximately twice the frequency of the deleterious allele. Thus, in previous models, the relative fitness of diploids vs. haploids is
Diploids thus have a mean fitness advantage over haploids when mutations are recessive (h < ½). Thus diploidy is advantageous with respect to mean fitness because of the masking of deleterious mutations (see Perrotet al. 1991; Otto and Goldstein 1992; Jenkins and Kirkpatrick 1994, 1995; Otto 1994; Otto and Marks 1996). Note that in the additive case (h = ½), the mean fitness of haploids and diploids is equal.
In the model presented here, mutation occurs during meiosis. Mutation thus occurs prior to selection in haploids and after selection, prior to mating, in diploids (Figure 2). For this reason, haploids entering selection have a higher frequency of the deleterious allele than diploids entering selection. This difference is reflected in the mean fitness of haploids vs. diploids as seen in Equation 4. By comparing Equations 4 and 6, it is clear that diploids can have a mean fitness advantage over haploids, even in situations where the deleterious allele is partially dominant (h > ½), and this is seen in Figure 3. Meiotic mutation thus causes diploidy to be favored over a larger range than seen in previous models because the frequency of the deleterious allele in haploids entering selection is greater than in diploids.
The difference in the frequency of the deleterious allele in haploids vs. diploids entering selection is affected by the resident level of diploidy in the population. In particular, if the population consists primarily of haploids, the equilibrium frequency (x̂_{2}) of the deleterious allele is small and new mutations arising during meiosis cause a large difference in the frequency of the deleterious allele in haploids vs. diploids. Thus diploidy is more likely to be favored in a population consisting primarily of haploids. In the region of the parameter space where polymorphic haploiddiploid life cycles are favored, diploidy has an advantage when rare, but not when common. This frequency dependence is the key attribute of the model that allows the evolution of polymorphic haploiddiploid life cycles. In previous models, the mean fitness of haploids vs. diploids did not exhibit frequency dependence.
Genetic associations: As seen in previous studies (Perrotet al. 1991; Bengtsson 1992; Otto and Goldstein 1992; Bell 1994; Otto 1994; Otto and Marks 1996), associations that arise during invasion of an allele modifying ploidy favor the evolution of haploidy. The positive association between modifiers that increase haploidy and the favored viability allele (Equation A2) arises through selection. Modifiers that increase haploidy are predominant in the haploid part of the population. After selection, the frequency of the favored viability allele in haploids is relatively high due to efficient removal of the deleterious allele by selection. Thus modifiers that increase haploidy end up with a purged genome that favors their invasion. The association between the modifier and viability loci becomes larger as linkage tightens, which favors haploidy to a greater extent (see Figure 4).
Meiotic mutation: The model presented requires the assumption of mutation linked to meiosis. Several lines of evidence suggest this is a reasonable assumption. There are data from mice that allow the mutation rate to be estimated for the perigametic interval, which is the period following the last mitotic division in the germ line and the first mitotic division in the zygote and thus includes meiosis as a major component (reviewed in Russell and Russell 1996; Russell 1999). These data indicate that the perigametic interval is highly mutagenic, at least for some loci. In particular, perhaps as many as 50% of all mutations arise during the perigametic interval in mice (Russell and Russell 1996). Since meiosis is the main event that occurs during the perigametic interval, these data suggest that meiosis is highly mutagenic. In addition, the observation that the per generation mutation rates for Drosophila, mouse, and human are similar has led to the hypothesis that a large fraction of mutations occur during meiosis, which occurs once per generation as opposed to during germ line mitotic divisions, which differ in number among these organisms (Russell 1999). Finally, some types of mutation are expected to occur more frequently during meiosis. For example, mutations that involve unequal crossing over (causing deletions and duplications), intrachromosomal crossing over (causing deletions and inversions), and nonhomologous interchromosomal crossing over (causing reciprocal translocations) are much more likely to occur during meiosis when recombinational machinery is active.
Acknowledgments
I thank M. Uyenoyama and M. Kirkpatrick for critical review during various stages of this project. I also thank A. Kondrashov for bringing my attention to the mouse perigametic mutation literature.
APPENDIX
The linearized recursion equations can be written in matrix form as v′ = Mv, where M is the 2 × 2 transformation matrix, v is the column vector {x_{3}, x_{4}}^{T}, and v′ is the same vector in the next generation. Denote v_{n} = {p,D}^{T} as a vector in the new basis, where p is equal to the frequency of the introduced modifier allele C_{2} (= x_{3} + x_{4}) and D is the standard measure of linkage disequilibrium (= x_{1}x_{4} – x_{2}x_{3}). The recursion equations in the new basis can be written in matrix form as
Define a vector v_{g} = {p*, D*} in the new basis such that the vector (I – N)v_{g} has its first entry equal to the characteristic equation evaluated at 1, and its second entry equal to zero. If λ = 1 were an eigenvalue of N, then all of the entries of (I – N)v_{g} would be zero, and v_{g} would be a right eigenvector of N. If z is a row vector of ones, then z(v_{g} – v_{g}′) = z(I – N)v_{g} = z
{C(1), 0}^{T} = C(1) and thus the condition for invasion under weak selection, C(1) < 0, is equivalent to z (v_{g} – v_{g}′) < 0. Thus the behavior of the system over one generation, when started at v_{g}, gives the asymptotic behavior of the system in the neighborhood of the C_{1} fixation. D* is thus the asymptotic disequilibrium that builds up between the viability and modifier loci upon introduction of a new modifier allele. The matrix equation (I – N)v_{g} can be written as
Footnotes

Communicating editor: M. Slatkin
 Received April 19, 2000.
 Accepted June 13, 2000.
 Copyright © 2000 by the Genetics Society of America