MATERIALS AND METHODS

The model: General models for the evolution of androdioecy must include the selfing rate, fecundity through hermaphroditic and male fertilization, and the frequencies of the two classes of individuals (e.g., Ross and Weir 1976; Charlesworth 1984). Hedgecock (1976) provides a model of male frequency change in C. elegans, but his analysis requires a number of equilibrium assumptions (e.g., equality of the reproductive potentials of males and hermaphrodites) that limit its applicability in the current case. Otto et al. (1993) developed a comprehensive model for the slightly more complex case of autosomal sex determination. Differences in C. elegans include hemizygous sex determination and production of males directly from hermaphrodites via chromosomal nondisjunction. Modifying the Otto et al. (1993) model using the mating table shown in Table 1, the change in male frequency (m) over time is given by m′=(1−σ)(1+2u)αm∕2T+β(1−δ)(1−αm)u∕T, (1) where T = αm[1 − σ(1/2 + u)] +β(1 − δ)(1 − σu)(1 − αm), α is equal to the fertilization success of males (αm is the proportion of female eggs that are outcrossed, 0 ≤ αm ≤ 1), β is equal to the proportion of eggs not fertilized by males that are self-fertilized, σ is equal to the relative viability difference between males and hermaphrodites, δ is equal to the degree of inbreeding depression in selfed offspring, and u is the rate of nondisjunction at the X chromosome (cf. Ottoet al. 1993, Equations 1 and 2). The existence of nondisjunction ensures that the males will always be maintained in the population at least on the order of the rate of nondisjunction (u; Chasnov and Chow 2002; A. Cutter, personal communication). Since u is known to be very low, it is easier to examine potential equilibria at higher frequencies by assuming that the contribution by nondisjunction to these equilibria is negligible. Under this parameterization, the equilibrium condition for the maintenance of androdioecy, α(1−σ)>2β(1−δ), (2) is the same in both the autosomal and hemizygous models (Ottoet al. 1993, Equation 4). Thus, depending on the degree of inbreeding depression, for males to be maintained other than at a nondisjunction-selection equilibrium they have to be significantly better at fertilizing eggs than hermaphrodites. Since inbreeding depression is thought to be fairly minor in C. elegans (Johnson and Wood 1982; Chasnov and Chow 2002), most of the influence on the equilibrium is likely to be through the relative fertilization success of males vs. hermaphrodites.

Equation 2 is very useful for illustrating the underlying factors that are important in maintaining males. If the parameters are allowed to become frequency dependent, however, potential solutions become more complex (Ottoet al. 1993). It is clear that the selfing rate in hermaphrodites must in general be frequency dependent because when the frequency of males is low it is unlikely that the males will be able to mate with all of the available hermaphrodites. This is especially the case in a species like C. elegans, where hermaphroditic sperm production precedes egg production (Wood 1988). Hermaphrodites that remain unmated shortly after reaching sexual maturity are therefore very likely to produce selfed offspring. As the frequency of males increases, the encounter rate between males and hermaphrodites should also increase, driving down the effective selfing rate. In C. elegans, however, the timing of sperm production makes it likely that there will always be some residual selfing regardless of male frequency, if only because some hermaphrodites go temporarily unmated just by chance. Some of the frequency dependence of this situation is encompassed in the above model since the selfing rate is itself frequency dependent [β(1 − m)(1 −αm); Table 1]. The underlying parameters themselves may also be frequency dependent, however. In particular, male fertilization success (α) may strongly depend on male frequency. For instance, mating success may be higher when males are rare than when they are common, especially because of male-male interference.

In the hemizygous model, we can solve for the frequency-dependent dynamics for male frequencies greater than the nondisjunction rate directly by noting that any male offspring in the population must be the result of a male-hermaphrodite cross. In the absence of meiotic drive (see LaMunyon and Ward 1997) and barring rare nondisjunction events, one-half of the offspring of a male-hermaphrodite cross will be males, and the frequency change of males across generations is given by m′=(1−s[m])∕2, (3) where s[m] is the frequency-dependent hermaphroditic “selfing rate” (the fraction of all fertilized eggs that are self-fertilized). Here s[m] is a composite of the mating and viability parameters given above, generalized to be frequency dependent. Rather than attempt to fully parameterize this model in terms of its component parts, we use a graphical approach that is well suited for empirical estimation.

The selfing function s[m] can in principle take any form, although it is constrained by the condition that s[0] = 1. For example, one trivial equilibrium can occur when the selfing rate approaches 0 as the male frequency becomes high (s[m] → 0 as m → ½), which results in a stable male frequency of m = ½, or what would effectively be a dioecious population. In practice, however, a great number of stable and unstable equilibria are possible depending on the form of s[m]. Rather than working directly with s[m], we have found it easier to use the entire left-hand side of Equation 3, which describes male frequency in the next generation as a function of the male frequency in the current generation. We call this the male maintenance function (MMF), because the male/hermaphrodite dynamics and equilibria can be described by the relationship between this function and the line of equilibrium, m′ = m. This general approach allows all of the potential influences on selection for or against males, including nondisjunction, to be determined empirically.

The MMF behaves in a very similar manner to a Ricker recruitment curve or return function used to describe population size dynamics (May and Oster 1976; Murray 1993). As shown in Figure 1A (using a complex curve chosen for illustrative purposes only), any time m′ goes above the equilibrium line, the population will evolve back toward it in the direction of increasing frequency, whereas when m′ goes below the line, the population will evolve back toward it in the direction of decreasing frequency. Thus, points where the MMF crosses the line starting above and ending below will be stable equilibria, while crossing points starting below and ending above will be unstable equilibria (Figure 1). Equation 1 yields MMFs with a single inflection point, which will either generate the Equation 2 equilibrium or not, depending on whether the curve starts above or below the equilibrium line. Equation 3 and the graphical analysis allows for potentially more complex equilibrium relationships.

MMF measurement: Estimation of the MMF requires measuring the change in male frequency from one generation to the next. Four male-enriched populations of the N2 (Bristol) strain of C. elegans were each started from three hermaphrodites and 20 males and grown on 6-cm nematode growth medium (NGM) plates seeded with Escherichia coli strain OP50 (Brenner 1974). From these plates, large numbers of eggs (>200) were selected at random and “picked” with a platinum wire onto five or six seeded 6-cm plates. Maintained at 20°, the worms were then sexed at the L4 stage and allowed to lay eggs. The sex ratio in this population served as an estimate of male frequency in the “parental” population (m). To avoid food depletion and to maintain constant density, the worms from each plate were suspended in ~1 ml of M9 solution, and a 300-μl fraction of this solution was transferred to a fresh plate when the progeny reached the L2 stage. The offspring developed to the L4 stage and were sexed, providing an estimate of male frequency in the “offspring” population (m′). This procedure was repeated 40 times using the same male-enriched population over a period of 4 weeks, allowing a wide variety of initial male frequencies to be sampled. Approximately 400 individuals per replicate were assayed for male frequency in two independent estimates, in both the parental and offspring generations, yielding a total of 70,753 individuals assayed.

Because we had no a priori expectation as to the shape of the response curve, we used the best-fitting polynomial derived from a stepwise regression to fit the data (Sokal and Rohlf 1994).

Male and fog-2 frequency assessment: Two male-enriched populations of the N2 strain of C. elegans were prepared as above. From these matings 150 eggs were selected at random, picked onto each of five 10-cm seeded NGM agar plates, and maintained at 20°. After maturing to the L4 stage, all individuals were sexed and allowed to lay eggs. Their progeny, likewise, matured to the L4 stage and were sexed. Because of the large number of worms (~1500/plate), all subsequent sexing was done using a cross-secting grid representing ~20% of the total plate area. To avoid food depletion and to keep the population size constant at ~1500 worms/plate, the worms from each plate were suspended in ~1 ml of M9 solution and a fraction of this solution (between 200 and 350 μl) was transferred to a fresh plate. The worms were sexed and transferred 6 of every 7 days. During transfers, all five lines were kept independent. The lines were maintained for 54 days, or ~15 generations.

For the fog-2 competition assays, five independent lines were established and maintained for ~15 generations as above except that eggs from hermaphroditic N2 and dioecious fog-2 populations were combined at the outset (50 N2 and 100 fog-2 eggs, i.e., 33% of each sex), creating what may be called a “trioecious” population. Female fog-2 individuals lay only one unfertilized egg every 8 hr if they are not mated (T. Schedl, personal communication, 1997), with the great excess of unlaid maturing oocytes accumulating in the ovaries. As more oocytes mature, they visibly compact against each other until they resemble the keys of a piano. This “piano effect” allows the fog-2 gene frequency to be assessed using offspring production and self-fertility criteria as described in Figure 2. The fog-2 populations used below were derived from an N2 (wild-type) background and backcrossed into N2 for eight generations (Schedl and Kimble 1988). These stocks were kindly provided by Tim Schedl (Washington University, St. Louis).

• Download figure
• Open in new tab
• Download powerpoint

Figure 1.

Male maintenance function for an androdioecious population. (A) A schematic representation of male frequency dynamics as theoretically predicted by the MMF. The dotted line gives the line of equilibrium at which there is no change in male frequency from one generation to the next (m′ = m). If the MMF falls above this line, then the population will evolve toward points of higher male frequency (i.e., male frequency is predicted to increase each generation). When the MMF falls below the equilibrium line, then the population evolves toward points of lower frequency. Stable and unstable equilibrium points are both possible, depending on the orientation of the MMF as it crosses the equilibrium line. This curve was chosen for illustrative purposes only and is not based on any theoretical prediction. (B) Experimental assessment of the MMF in C. elegans. The ability of males to outcross is observed (open diamonds) to vary with male frequency. Given an initial male frequency, m, the quartic function fit to the observed data predicts the frequency of males in the offspring, m′ (solid curve). Also depicted is the 95% confidence interval for the quartic fit curve (dotted curves). The equation for the quartic curve is given as m₁ = β₁m + β₂m² β₃m³ + β₄m⁴ + ε with the y-intercept set to 0 (Table 2). For male frequency to remain constant, the quartic curve must cross the maintenance line (dashed line). Any point where such a crossing exists (0, 0.14, 0.24) is predicted to be an equilibrium point, although the confidence region includes many points on either side of the maintenance line. The least-squares best fit of the model with frequency-independent parameters (Equation 1) is shown as the boldface dashed line (α = 0.52, β = 0.2, σ = 0, δ = 0, u = 0.002).

• Download figure
• Open in new tab
• Download powerpoint

Figure 2.

Schematic overview of the fog-2 gene assay. Young larval individuals were placed into wells. If they developed male morphology (fans) or had hermaphroditic morphology but laid no eggs (piano effect), they were classified, respectively, as males or females. True hermaphrodites laid eggs and were classified as homozygous wild type or heterozygous by whether their offspring all laid eggs (homozygotes) or some did not (heterozygotes).

The count data were analyzed using a logistic regression approach with the CATMOD procedure of SAS (SAS Institute 1988). The model used is analogous to an analysis of covariance using categorical data (Hagenaars 1990). A completely factorial model was used with line as a main effect and day as a logistic covariate. Residuals displayed no significant serial autocorrelation, so no autocovariance function was fitted to the data (Chatfield 1984). Roughly 3000 individuals were counted per day across the five lines for a total of 112,977 and 193,512 individuals sexed in the androdioecious and fog-2 assays, respectively. Parameter solutions and significance tests were obtained using an iterative maximum-likelihood approach (SAS Institute 1988).