Monte Carlo Studies of Plant Mating System Mixed Mating Models Estimation Models: the One-pollen Parent And

Estimation of mating system parameters in plant populations typically employs family-structured samples of progeny genotypes. These estimation models postulate a mixture of self-fertilization and random outcrossing. One assumption of such models concerns the distribution of pollen genotypes among eggs within single maternal families. Previous applications of the mixed mating model to mating system estimation have assumed that pollen genotypes are sampled randomly from the total population in forming outcrossed progeny within families. In contrast, the one-pollen parent model assumes that outcrossed progeny within a family share a single-pollen parent genotype. Monte Carlo simulations of family-structured sampling were carried out to examine the consequences of violations of the different assumptions of the two models regarding the distribution of pollen genotypes among eggs. When these assumptions are violated, estimates of mating system parameters may be significantly different from their true values and may exhibit distributions which depart from normality. Monte Carlo metho d s were also used to examine the utility of the bootstrap resampling algorithm for estimating the variances of mating system parameters. The bootstrap method gives variance estimates that approximate empirically determined values. When applied to data from two plant populations which differ in pollen genotype distributions within families, the two estimation procedures exhibit the same behavior as that seen with the simulated data. HE mating system of a population determines how genetic information is T transmitted between generations. Knowledge of the mating system is central to understanding the mechanisms of evolutionary change. One approach to describing the mating system in plant populations has involved the use of genetic markers to reconstruct the pattern of mating through analysis of parental and progeny genotypes (FYFE and BAILEY 1951; BROWN and ALLARD 1970; CLEGG, KAHLER and ALLARD 1978; CLEGG 1980; SHAW, KAHLER and ALLARD 1981; RITLAND and JAIN 1981; SCHOEN and CLECC 1984). This approach entails the estimation of parameters which underlie models of plant mating systems. The parameters of interest include the proportion of seeds produced via random outcrossing as opposed to self-fertilization, the frequenGenetics 114: 927-945 April, 1986 928 D. J. SCHOEN AND M. T. CLEGG cies of alleles in the outcrossed pollen pool (the pollen involved in outcrossing events) and the genotype frequencies of the mating parents. Mating system estimation is relevant to many areas of evolutionary and applied genetics. For example, the effect of the mating system on genotype frequencies must be known in order to estimate the direction and intensity of selection acting on genotypes (CLEGG 1983). Knowledge of the mating system has also proven useful in understanding the structure of genetic variation among populations (LEVIN 1977; BROWN 1979; LOVELESS and HAMRICK 1984), the level of within-population genetic variability (BROWN 1979; HAMRICK, LINHART and MITTON 1979) and the response of populations to selection (ALLARD 1960). In estimating mating system parameters it is typically necessary to make several assumptions about the mating process. The assumptions most widely used for this purpose are incorporated in the mixed mating model (HAYMAN 1953; CLECC 1980). They state that (1) mating events are of two typesrandom outcrossing (occurring with probability t) and self-pollination (occurring with probability s = 1 t ) ; (2) outcrossing rates are constant and independent of maternal genotype; and (3) pollen genotypes are distributed uniformly among all eggs (i.e., each outcrossing event can be viewed as resulting from a random draw of pollen from an infinite pool of pollen of all genotypes). There are several reasons for choosing the mixed mating model to represent the pattern of genetic transmission in plant populations. First, many plants are self-compatible and may be self-pollinated to some degree. Second, the mixed mating model provides a simple description of the mating system, requiring the estimation of only the outcrossing rate and pollen pool allele frequencies. Third, the mating systems of many plant populations may be reasonably approximated by the assumptions of the mixed mating model. The mixed mating model has often been successfully applied to mating system estimation in plants (CLEGG 1980). Nevertheless, in certain cases the mixed mating model fails to provide an adequate description of the mating system. For example, ENNOS (1 98 1) found that heterozygous maternal parents in populations of Ipomoea species produced significantly more heterozygous offspring than expected under the mixed mating model. Such a heterozygous excess can arise due to violation of the mixed mating model assumption of a uniform pollen genotype distribution among eggs, as may occur in certain insect-pollinated species when pollen is picked up from one or a few previously visited flowers and is deposited en masse on the stigma (SCHOEN and CLECG 1984). In an earlier publication, we specified an alternative model (the one-pollen parent model) that provides improved mating system parameter estimates when the pollen involved in the cross-fertilization of the eggs of a maternal parent derives only from a single paternal parent genotype (SCHOEN and CLECC 1984). In this paper, we examine in detail the performance of both models, using Monte Carlo simulation. We first ask what is the distribution of mating system parameter estimates obtained from both the one-pollen parent and mixed mating models, and how do estimates and true population values compare when there is violation of assumptions regarding the sampling of pollen genotypes within MATING SYSTEM PARAMETERS 929 maternal families. This question is posed for a much larger number of parameter combinations than in our earlier paper, and the results obtained are often qualitatively different from those previously reported by us, especially for asymmetrical gene frequencies. Next, we examine the accuracy of variance estimates of mating system parameters obtained through the computationally simple bootstrap algorithm (EFRON 1979). This is done for simulations in which the pollen genotype distribution assumptions either are or are not violated. Lastly, we investigate each model’s performance with data from two plant populations. The populations studied are especially appropriate test cases, as independent evidence has indicated that they differ with regard to the sampling of pollen genotypes within maternal families (SCHOEN 1985). MATERIALS AND METHODS The simulation algorithms assume an infinitely large population with a mating system characterized by the parameter values we seek to estimate (referred to as “true” values of mating system parameters). The simulations yield an array of progeny genotypes organized by family. For simplicity, we assume a single diallelic marker locus. Two methods of simulating family-structured data were used. In the first method, the probability of an egg being outcrossed with pollen of genotype A I is equal to the product of the frequency of the allele in the pollen pool and the outcrossing rate, p t . In the second, when an egg is outcrossed with pollen of a particular paternal genotype, all other outcrossing events of that seed parent involve the same paternal genotype. Frequencies of parental genotypes in both simulation methods are based on inbreeding equilibrium. The two simulation methods correspond with assumptions made in the mixed mating model and one-pollen parent model, respectively, and are hereafter referred to as the mixed mating model and one-pollen parent model simulation methods. Each of two estimation procedures were applied to the simulated family-structured data; i.e., the estimation procedure based on the mixed mating model (CLEGG, KAHLER and ALLARD 1978) and the procedure based on the one-pollen parent model (SCHOEN and CLEGG 1984). When employing each procedure, we assume that maternal parent genotypes (or mating types) are unknown, and hence, estimation involves inferring a 3 X 3 (or 9 X 3) array of the numbers of progeny genotypes produced by the different maternal parent genotypes (or mating types) (CLEGG, KAHLER and ALLARD 1978; SCHOEN and CLEGG 1985). This assumption is often necessary because ageor tissuespecific expression of allozyme phenotypes (BROWN and ALLARD 1970) may lead to errors of identification of the maternal genotype. The procedure based on the mixed mating model gives estimates of the outcrossing rate ( t ) , the frequency of the A I pollen in the pollen pool ( p ) and the frequency of the genotypes A I A I , A1A2 and A2A2 among the maternal parents ( M I , , MI2 and MZ2). The procedure based on the one-pollen parent model gives estimates of the paternal parent genotype frequencies ( P I ] , PI* and P22) in place of the pollen pool allele frequency. Table 1 shows the transition probabilities for both estimation models. The maximum likelihood and gene counting estimation procedures as applied to mating system estimation by CLEGG, KAHLER and ALLARD (1978) and SCHOEN and CLEGG (1984) were used. In all above-described simulations, five progeny per family (k = 5) and 200 families (n = 200) were sampled. Sampling of families was simulated for populations characterized by the nine possible pairwise combinations o f t = 0.8, 0.5 and 0.2 and p = 0.5, 0.6 and 0.8. Bootstrap estimates (EFRON 1979) of standard deviations for all mating system parameters were obtained using the following algorithm. A simulated data set, produced as described above (but with k = 5 and TI = 50) was randomly resampled with replacement to yield a “bootstrap” sample of n = 50 families. After a bootstrap sample was produced, it was used to obtain estimates of mating system parameters. The process was carried 930 D. J. SCHOEN AND M. T. CLECG TABLE 1 Probabilities of observing progeny genotypes given maternal parent genotypes (mixed mating model) or mating types (one-pollen parent model)