Abstract
The standard approaches to estimation of quantitative genetic parameters and prediction of response to selection on quantitative traits are based on theory derived for populations undergoing random mating. Many studies demonstrate, however, that mating systems in natural populations often involve inbreeding in various degrees (i.e., self matings and matings between relatives). Here we apply theory developed for estimating quantitative genetic parameters for partially inbreeding populations to a population of Nemophila menziesii recently obtained from nature and experimentally inbred. Two measures of overall plant size and two of floral size expressed highly significant inbreeding depression. Of three dominance components of phenotypic variance that are defined under partial inbreeding, one was found to contribute significantly to phenotypic variance in flower size and flowering time, while the remaining two components contributed only negligibly to variation in each of the five traits considered. Computer simulations investigating selection response under the more complete genetic model for populations undergoing mixed mating indicate that, for parameter values estimated in this study, selection response can be substantially slowed relative to predictions for a random mating population. Moreover, inbreeding depression alone does not generally account for the reduction in selection response.
WIDESPREAD interest in assessing the potential for response to either artificial or natural selection has motivated numerous studies of quantitative genetic variation within populations of plants and animals. These studies have generally employed experimental designs in which traits are measured on progeny obtained from controlled crosses, where parents are chosen and assigned mates at random (Comstock and Robinson 1948). Components of phenotypic variance obtained from such approaches are therefore understood to refer to a random mating population, as are the resulting predictions of selection response. Extensions of these approaches accommodate assortative mating, i.e., mate pairing according to phenotypic similarity with respect to a particular trait (Crow and Kimura 1970; Falconer and Mackay 1996), a mating scheme that corresponds in some cases to the mating pattern in particular populations of interest (e.g., human; Fisher 1918) and that tends to increase statistical power and precision of estimates of genetic components.
Investigations of mating systems of numerous wild populations have, however, indicated that deviations from random mating are the rule and that inbreeding is common (reviewed in Schemske and Lande 1985; Thornhill 1993; Waller 1986). Mating systems can, in principle, encompass a wide range of relationships between mating individuals (see Waser 1993, pp. 14) and thus lead to broad variation in the degree of inbreeding of the resulting offspring. Exhaustive inference of the relationships between all mating pairs has not been possible, but variable inbreeding (i.e., “mixed mating systems”), resulting in intermediate average rates of inbreeding, has now been documented for a wide variety of species, especially plants, and interpopulation variation in rates of inbreeding is also well established (see Table 1 for recent estimates).
These findings of substantial rates of inbreeding in many taxa have motivated studies to assess consequences of inbreeding. Many of these focus on detecting and quantifying inbreeding depression, the rate of reduction in the mean value of a trait in a population relative to the increase in the degree of inbreeding. Inbreeding depression has long been recognized (e.g., Falconer and Mackay 1996, Table 14.1) and has now been documented for many taxonomic groups (e.g., reviews in Thornhill 1993). Theory predicting effects of mating systems on degree of inbreeding depression and the reverse (e.g., Lande and Schemske 1985; Charlesworth and Charlesworth 1987; Uyenoyamaet al. 1992; Waller 1993; Schultz and Willis 1995) has prompted further empirical investigation of the relationship between the degree of inbreeding and inbreeding depression (e.g., Latteret al. 1995; Doumset al. 1996; Husband and Schemske 1996; Johnston and Schoen 1996; Mayeret al. 1996; Ballou 1997). Likewise, predictions of effects of mating system on genetic variability have stimulated empirical study of relationships between them (reviewed in Charlesworth and Charlesworth 1995).
The common finding of partial inbreeding in natural populations suggests that quantitative genetic predictions of R, the pergeneration response to selection, may often be misleading when, as is typical, they use the breeder’s equation,
These methods for partitioning genetic variation of quantitative traits in inbreeding populations paved the way for prediction of selection response with inbreeding, and interest in the interplay between selection and inbreeding has been growing over the past decade. Wright and Cockerham (1985) and Meuwissen and Woolliams (1994) developed expressions for predicting shortterm responses to selection. Recently, this work has been considerably extended by Kelly (1998), who has developed recursions for additive and inbred dominance variance components in mixed mating populations undergoing selection. On the basis of extensive simulation studies, de Boer and colleagues concluded that the variance components and their random effects associated with homozygous dominance can, given the relatively slight inbreeding in many livestock populations, be safely ignored in the prediction of genetic merit in livestock (de Boer and van Arendonk 1992; de Boer and Hoeschele 1993).
Considering the converse effects of selection on rates of inbreeding, Wray and Thompson (1990) developed theory for predicting inbreeding in finite populations undergoing selection (see also Woolliamset al. 1993). Methods for maximizing response to artificial selection while restricting increase in inbreeding have also been compared (Toro and PerezEnciso 1990; Brisbane and Gibson 1995). To predict selection response over many generations, it is necessary to take into account genetic processes in addition to inbreeding, including drift, linkage disequilibrium, and the input of variation by spontaneous mutation, as accomplished by Wei et al. (1996).
Despite these developments, little empirical work has taken advantage of theory addressing the effects of inbreeding on the structure of quantitative genetic variation and on response to selection. Two such studies concern domesticated populations (maize, Cornelius 1988; and sheep, Shaw and Woolliams, 1999). Here, we present a study of an annual plant, Nemophila menziesii, experimentally inbred from a random sample of a wild population. We measured pedigreed plants that differ in degree of inbreeding for morphological traits, as well as time of flowering. We partitioned phenotypic variation of these traits according to models that account for inbreeding as summarized below. We demonstrate that genetic components of variance specific to inbreeding populations make substantial contributions to phenotypic variance in our inbred population of N. menziesii. Simulations based on our estimates indicate that evolution in partially inbreeding populations, even over few generations, can differ strongly from predictions employing narrowsense heritability alone, as well as predictions corrected to allow for inbreeding depression.
THEORY, MATERIALS AND METHODS
Theory: Considering effects on a trait, y, measured in an individual bearing alleles i and j at a single segregating locus, we can use a general model to separate genetic effects from effects due to environmental conditions:
In a random mating population, HardyWeinberg frequencies ensure that
Compared to a random mating population, inbreeding increases the frequencies of homozygous genotypes, or more specifically, of autozygous genotypes, in which paris of homologous alleles are IBD. In this case, the expectation of the dominance deviations does not remain zero. The expected trait value E(y) of an individual inbred to degree F then includes inbreeding depression μ_{F}, i.e., the (nonzero) expectation of the autozygous dominance effects, expressed in inbreds,
Apart from its widely recognized effect on the population mean, inbreeding alters the population variance, in part because inbreeding destroys the simplifications given in Equations 2 and also because the variance of autozygous dominance deviations may differ from the variance of dominance deviations under random mating (V_{D}, above). In the framework developed by Harris (1964) and Weir and Cockerham (1977), the variance associated with autozygous dominance comprises three distinct components:
V_{DI} = E(d^{2}_{ii})  (E(d_{ii}))^{2}, the total variance due to autozygous dominance effects. This is the dominance variance of a completely inbred population.
H^{*} = (E(d_{ii}))^{2}, the squared perlocus inbreeding depression, summed over loci.
Cov(a, d_{I}) = E(a_{i}d_{ii}), the covariance between the additive effect of alleles and their autozygous dominance deviations.
Thus, the variance of autozygous dominance deviations can differ from that of “random” dominance deviations (i.e., dominance deviations attributable to allelic combinations that are not IBD). To emphasize this, we relabel V_{D}, the dominance variance in a randomly mating population, to V_{DR}. Using these definitions, the variance of individual phenotypes can be written as
Study species: N. menziessi is a selfcompatible annual plant of the family Hydrophyllaceae. It is native to California and Oregon. Munz (1959) described the species as a “very variable complex.” Within Riverside and San Bernardino Counties, differences in plant stature, leaf morphology, and flower color and size distinguish mountain populations from those below 800 m, and these differences are maintained under common conditions (R. G. Shaw, G. A. J. Platenkamp and R. H. Podolsky, unpublished data). The flowers bear nectar and tend to attract diverse pollinating insects (Cruden 1972; Andersson 1994). They are also protandrous, a trait that allows removal of pollen from a given flower before its stigma is receptive. Thus, the potential for avoidance of autogamy is great. However, in the population we studied, unvisited flowers readily selfpollinate, and these flowers produce viable seeds (R. G. Shaw and D. L. Byers, personal observations). Moreover, seeds have no specialized dispersal mechanism or apparent means of longdistance dispersal. These aspects of the plant’s biology indicate that inbred matings, whether by selfing or by mating between close relatives, may commonly contribute offspring to succeeding generations in the population under consideration. In a field study of several mountain populations of N. menziesii, C. T. Schick (personal communication) estimated selfing rates as high as 28%, even in populations that failed to set fruit in the absence of pollinators. To assess the potential genetic and evolutionary impact of partial inbreeding in this population, we experimentally inbred plants and assayed the following traits: two measures of overall plant size, height, and number of nodes at first flower, two measures of floral size, petal length and width, and days to first flower.
Genetic design and trait assays: Estimation of the components of genetic variance that arise with inbreeding requires observations on groups of individuals in which pairs can be predicted (from the pedigree) to share alleles in autozygous form (thus contributing information for V_{DI} and H^{*}) or with one individual autozygous and the other heterozygous [contributing information for Cov(a, d_{I}); Cockerham and Weir 1984]. On the basis of a simulation study (F. H. Shaw, unpublished results; see also Cockerham and Weir 1984), the crossing scheme (Figure 1) was judged suitable for estimating the variance components that contribute to phenotypic variance under inbreeding, in addition to the “random” genetic components.
To obtain parents for the initial crosses, seedlings were collected in January 1990 from an uncultivated area of the University of California at Riverside (UCR) Botanic Gardens. Seedlings were sampled at 2m intervals along parallel transects 2 m apart to reduce the chance of sampling close relatives. These plants were grown to maturity in a greenhouse at UCR. A total of 52 plants chosen at random served as paternal parents (sires) and 156 as maternal parents (dams) in a nested crossing design (a distinct set of 3 dams crossed with each sire). The progeny of these crosses, termed generation 1, are considered the reference generation, with inbreeding coefficients (F) of zero [Shaw et al. (1995) report genetic variation in response to biotic conditions in a field experiment using progeny of these crosses]. Plants of generation 1 used in this study and all later generations were grown, crossed, and measured in a greenhouse at the University of Minnesota.
In January 1994, 40 of these 52 progeny groups (halfsibships, each comprising three fullsibships) were chosen at random as sources of parents in the next series of crosses; a distinct set of eight progeny groups was chosen at random for each of five crossing blocks (Figure 1). Within each block, five progeny groups were designated at random as sources of sires in further matings, while the remaining three progeny groups provided individuals to be used as dams. Individuals randomly chosen from these progeny groups were grown, and measures of petal length and width were obtained for each. Within each crossing block, plants were mated factorially, yielding progeny in generation 2 with F = 0 [a study based on these crosses is reported in Byers et al. (1997)]. In addition, all maternal plants were selfpollinated, the resulting progeny (also generation 2) having F = 0.5.
Assays of generation 2 and a further series of crosses were initiated in December 1994. Generation 2 was subsampled to establish 10 crossing blocks, 2 from each of the crossing blocks of the previous generation. In generation 2, each crossing block consisted of nine individuals descended from one pair of grandsires in the founding generation (Figure 1). This group was composed of three trios, each comprising a pair of fullsibs and a selfsib (maternal halfsib produced by selfing the maternal parent). These trios were chosen such that they had in common the maternal grandsire and also, in the case of the noninbred individuals, the sire. In addition, two plants within each trio (in Figure 1, plants 2 and 3, with F = 0 and F = 0.5, respectively) were each crossed to a randomly chosen plant that shared no ancestors in the known pedigree. Individuals chosen for the next series of crosses according to the above scheme, together with an additional five fullsibs of each, were grown and measured (see below). The crosses produced seeds with the following array of inbreeding coefficients: 0, 0.06, 0.25, 0.5, and 0.75. All crosses were carried out reciprocally. We refer to the offspring from this set of crosses as generation 3.
Assays of generation 3 and a further series of crosses were initiated in August 1995. Five plants from each set of progeny (e.g., AJ in Figure 1) were grown and measured. One individual from each fullsib group was selfed to produce progeny not considered further here.
Summarizing the available observations, petal length and width were measured on the 114 individuals representing generation 1, and five traits, date of first flower, size (height and number of nodes) at first flower, and petal length and width for the first opened flowers, were measured on 450 and 1226 individuals in generations 2 and 3, respectively. Single petals of each of 35 flowers were measured in millimeters with digital calipers. Height was measured with a meter rule in centimeters. Progeny from every level of inbreeding planned were measured, but not every lineage was represented fully according to the design given above, as a result of germination failure, mortality, or sterility (Table 2). To the extent that deviations from the intended design are due to selection, they are likely to bias estimates of inbreeding depression (i.e., indicating weaker inbreeding depression than the actual), but to have negligible influence on maximum likelihood estimates of the variance components, according to a simulation study of Shaw and Woolliams (1999).
Analysis: Restricted maximum likelihood (REML) was used to estimate the parameters of the full model (Equations 3 and 4) of trait determination and to test hypotheses (Shaw 1987; Shaw and Woolliams 1999). Quantile plots of residuals confirmed the validity of the normality assumption of the REML analysis. Because all crosses were done reciprocally, variance due to maternal effect could readily be distinguished from the genetic components defined in Equation 4. Thus, all models also included a random maternal effect with variance V_{M}. For simplicity, maternal effects were treated as perfectly correlated between individuals having the same maternal parent. They were thus modeled as arising either from environmental effects unique to maternal individuals or from cytoplasmic genetic effects. To estimate inbreeding depression as the linear change in the trait mean in response to degree of inbreeding, F of each individual was included as a linear covariate in all analyses. Estimation of inbreeding depression in this way appropriately accounts for the lack of independence between individuals linked through the pedigree structure (see Lynch and Walsh 1998, pp. 262265, for discussion of this issue). A fixed effect for the block (spatial and temporal) in which each plant was grown was also included in each analysis. In preliminary analyses, tests of the effect of maternal inbreeding were also conducted by including maternal F as a covariate. However, effects of maternal inbreeding were consistently found to be weak and not significant (in contrast to Hauser and Loeschke 1996), and these are not considered further.
Likelihood ratios (Kendall and Stuart 1973) were used for hypothesis testing. The null hypothesis that Cov(a, d_{I}) = 0 was tested first, by eliminating that parameter and comparing the likelihood of that reduced model to the likelihood of the full model. V_{A} and V_{DI} were tested by a similar procedure, except that V_{A} = 0 or V_{DI} = 0 requires that Cov(a, d_{I}) = 0. Thus, the covariance was omitted from the model for tests of these two variance components. Other components were tested against the full model, using 2.7 as the critical value for α = 0.05. This is appropriate when the null hypothesis coincides with the feasible limit for the parameter (i.e., variance components less than zero are not allowed; Self and Liang 1987). We also compared the variance of autozygous dominance deviations with that of random dominance deviations by testing the null hypothesis that V_{DI} equals V_{DR}. Standard errors of components of variance were obtained from diagonal elements of the asymptotic variancecovariance matrix of the maximum likelihood estimates (Searleet al. 1993; see appendix). These can be used for an approximate test of the parameters, but do not completely coincide with the more rigorous likelihood ratio test.
Estimates for the component H^{*} were all negative, but their substantial sampling variances and very small likelihood ratio test statistics suggest that these are attributable to sampling error. If inbreeding depression is due to the composite effects of many loci, H^{*}, which is the sum of the squares of these single locus effects, should be near zero. The component H^{*} is not reported in our analyses because it was invariably constrained to zero to satisfy feasibility.
Simulations: To assess the impact of the complete dominance model (Equations 3 and 4) and partial inbreeding on projected shortterm response to selection on a quantitative trait, we simulated finite populations undergoing five generations of selection on a single trait with genetic determination corresponding to that estimated for height and for petal width. Populations of size 100 were simulated as noninbred progeny of 20 unrelated founders. Each founder was assigned 2 unique alleles at each of 30 loci (1200 alleles in all). This contrasts with previous simulations of this model that use only two alleles per locus in different frequencies (de Boer and Hoeschele 1993; Kelly 1998). The biallelic approach has advantages (variance components can be easily calculated from allele frequencies; Cockerham and Weir 1984) and disadvantages (autozygosity occurs without inbreeding, and artificial dependencies varying with allelic frequencies exist between the variance components). Under the full genetic model, correlated values for the additive effect (a_{i}) and autozygous dominance deviation (d_{ii}) for each allele were sampled from
Distinct mating systems were simulated: random mating, 20% selfing, and 50% selfing (with the remaining matings at random in the latter two cases). In each case, 100 offspring were produced from the individuals in the mating pool for each generation. For five initial generations, transmission proceeded by the specified mating system, in the absence of selection. Thereafter, we imposed linear directional selection on the phenotype for five generations, according to the following scheme: individuals in generation k joined the mating pool with probability
RESULTS
Morphological traits: For each of the morphological traits, additive genetic variance (V_{A}) made highly significant contributions to the phenotypic variance. Narrowsense heritabilities of the traits (computed with V_{P} as for a random mating population, as the sum of V_{A}, V_{E}, V_{DR}, and V_{M}; Table 3) ranged from 16% for petal length to 26% for height, while additive genetic coefficients of variation, CV_{A}, ranged from 5% for petal length to 15% for node number. Highly significant inbreeding depression was also consistently found (Table 3), indicating that, with inbreeding, plants tended to decline in size with respect to each of these characters.
The estimates of variance components associated with dominance differed among traits far more strikingly. In the case of wholeplant size traits (plant height and node number), although inbreeding depression was detected as significant for both traits and the random component of dominance variance, V_{DR}, was significant for node number, V_{DI} appeared to be negligible. In the case of node number, the likelihood was maximized at a very small negative value for this variance component, indicating that the best estimate for both V_{DI} and Cov(a, d_{I}) is zero; under this constraint, the estimates of the remaining components and their standard errors differed little from the values given. Thus, for both overall size traits, the phenotypic variance was largely accounted for by the components, V_{A}, V_{E}, V_{DR}, and V_{M}.
For the petal size traits, V_{DR} of petal width was significant, but that for petal length, similar in magnitude but substantially smaller in relation to the remaining components, was not. The estimates of V_{DR} were substantially exceeded by the estimates for V_{DI}. These were up to four times as large as V_{DR} and contributed significantly to V_{P} (petal length, P < 0.025; petal width, P < 0.05), despite their large standard errors in the full model (Table 3). The differences between V_{DI} and V_{DR} were not detected as significant. The covariance between additive effects and autozygous dominance deviations [Cov(a, d_{I})] was estimated as positive for petal length and negative for petal width; in neither case was it significantly different from zero (P > 0.5 and 0.15, respectively). Thus, the additive effects of alleles associated with these traits appear to be weakly correlated with their autozygous dominance deviations.
Although the design had sufficient power to detect V_{DI} for floral size traits as significant, it tended to yield lower precision for estimates of V_{DI} and Cov(a, d_{I}) than for the other components of variation (Table 3). Moreover, the sampling covariance between these two parameters was substantial and negative (see appendix; e.g., sampling correlation was 0.76 and 0.83 for petal length and width, respectively), as was that between Cov(a, d_{I}) and V_{A} (sampling correlation was 0.61 and 0.73 for petal length and width). Thus, the inclusion in the model of Cov(a, d_{I}) can substantially affect the estimates of V_{A} and V_{DI}. Under a model omitting Cov(a, d_{I}), the estimates of V_{A} for petal length and width were 0.35 ± 0.11 and 0.17 ± 0.06, respectively, and the estimates of V_{DI} were 0.56 ± 0.33 and 0.25 ± 0.18, respectively.
Other contributions to variation in these morphological traits were also substantial. Differences between generations were detected (not shown), with plants in later generations tending to be larger in whole plant measures, but smaller in size of petals. These generation effects are distinct from effects of inbreeding. Maternal variance (V_{M}) was detected as highly significant for two traits, number of nodes and petal length, and accounted for 8 and 3% of the variance in those traits, respectively.
Flowering date: The effect of inbreeding on mean time to flowering was positive, indicating that more inbred plants tend to flower later, but this effect was not statistically significant. Considering estimates of genetic components of variance, both V_{A} and V_{DR} were substantial, with h^{2} computed as for a random mating population of ∼30% and CV_{A} of 12.3%. V_{DI} was extremely large, greatly surpassing V_{E}, and highly significant. In comparison, Cov(a, d_{I}) was relatively small (r_{a,}_{d}_{I} = 0.3) and did not differ significantly from zero. The weakness of both this covariance and the inbreeding depression for this trait suggests that dominance at the loci influencing this trait is not strongly directional. V_{M} accounted for ∼7% of the random mating phenotypic variance and was highly significant.
Simulations of selection response: Before the onset of selection, the simulated populations accumulate inbreeding approximately according to predictions accounting for the numbers of individuals (Falconer and Mackay 1996, p. 67; here, 20 founders and 100 individuals every generation thereafter). For example, in all random mating simulations, the average inbreeding before selection in generation 6 is 0.047 (not shown, prediction of 0.044). Selection increases the rate of accumulation of inbreeding (generation 10: F = 0.09 vs. prediction in the absence of selection of 0.07), as expected (e.g., Robertson 1961; Wray and Thompson 1990; Santiago and Caballero 1995). Predictions are likewise confirmed for the case of a random mating population with dominance contributions to the trait restricted to the random dominance (V_{DR}; i.e., Model 1). Here, the average response observed over 5 generations in our simulations closely approximates that given by Equation 1 (predictions: height, 0.6 cm; petal width, 0.48 mm); however, even under this simple model of genetic determination, partial selfing modifies the response to selection substantially, enhancing it in the case of 50% selfing by ∼5% per generation over the prediction based on the narrowsense heritability. This effect is expected, because inbreeding induces a correlation between the effects of the two alleles that individuals carry at a locus and, thus, increases the contribution of V_{A} to the phenotypic variance (see Equation 4). The increases in selection response that we observe under Model 1 with partial inbreeding are quantitatively consistent with this effect. In these and all other simulations, the realized selection differential (not shown) very closely matched the specified selection differential, deviating by <1% per generation for petal width and <2.3% for plant height. These slight differences between specified and realized selection differential do not account for deviations in selection response from predictions or for differences in response among the genetic models or mating schemes.
Inbreeding depression affecting the mean of the trait in opposition to the direction of selection (Model 2), as found for plant height and petal width, can dramatically slow response to selection in all three mating schemes. In the case of the random mating population, this reduction results from inbreeding caused by finite population size alone. With partial selfing, the population mean can quickly decline below its initial value and not regain it, despite gradual increases due to selection (e.g., Figure 2, b and c). The observed reductions in trait mean relative to means observed in the absence of inbreeding depression (Model 1) closely match the predictions from standard theory [Falconer and Mackay (1996, Chap. 14), i.e., the product of the mean coefficient of inbreeding and the inbreeding depression slope, 6.6 for height and 0.63 for petal width; Table 3]. Considering the case of 50% selfing, the mean F at generation 10 is 0.4, such that the expected reduction in mean for height is 2.7 and for petal width is 0.25.
Under the full model (Model 3), when the estimate of V_{DI} is substantial and that of Cov(a, d_{I}) is small and negative, as for petal width, selection response can be reduced substantially further, with somewhat greater reduction the higher the selfing rate (Figure 3). For plant height, where V_{DI} makes a far smaller contribution to variation, the reduction of selection response under Model 3, relative to Model 2, is relatively slight. The discrepancy between the responses under Models 2 and 3 is essentially eliminated when Cov(a, d_{I}) is absent (not shown; means for each generation coincide with those for Model 2 within 1.5%).
DISCUSSION
In keeping with many previous studies of inbreeding in diverse organisms (see Introduction), this work has demonstrated strong inbreeding depression in a population of N. menziesii recently collected from nature. Matings between more closely related individuals produced progeny that were smaller overall and bore smaller flowers. Beyond this, we have quantified the novel components of genetic variance expected to arise with inbreeding, V_{DI}, H^{*}, and Cov(a, d_{I}). Although the study gave no evidence that these components contribute to variation in two measures of plant size (height and number of nodes), it demonstrated that V_{DI} contributes substantially to genetic variation in three reproductive traits (petal length and width and flowering date).
In the petal size traits, significant inbreeding depression is accompanied by significant V_{DI}, although not by significant Cov(a, d_{I}). Thus, inbreeding increases the genetic variance for the trait, but this increase in variance is accompanied by a reduction in the mean, opposing selection favoring larger flower size. For both petal size traits, estimates of V_{DI} appreciably exceeded those for V_{DR}, although the difference was not detected as significant; in the case of petal length, V_{DR} was not significant. These findings may illuminate earlier ones in which V_{D} was not detected in progeny resulting from random mating, despite apparent inbreeding depression (e.g., of seed mass; Montalvo and Shaw 1994). Evidently, the component of dominance variance attributable to effects that contribute to inbreeding depression, V_{DI}, may, as in this case, be large even though V_{DR} is negligible.
A surprising pattern of genetic determination was found for the two size traits, height and node number at flowering. These traits showed clear inbreeding depression, yet no detectable homozygous dominance variance (V_{DI}). Similarly, Shaw and Woolliams (1999), in their study of sheep subjected to rapid inbreeding, could not detect significant V_{DI} or Cov (a, d_{I}) for live weight in a large pooled dataset of crossbred lambs although strong evidence of V_{DR} and inbreeding depression were found. In that case, the estimate of V_{DI} was similar in magnitude to that of V_{DR}. One possible explanation for these results is that the contribution of each of very many loci to inbreeding depression, although consistent in reducing the trait value, is individually extremely small, and thus the variance of the homozygous dominance deviations cannot be detected.
In only one of the traits studied, flowering time, was the estimate of inbreeding depression not significant. This trait exhibited substantial V_{DR} and V_{DI}, however. Taken together, these results indicate that, for this trait, the homozygous dominance deviations are not strongly directional. Inbreeding thus enhances the variance for flowering time without significantly affecting the mean.
Under partial inbreeding, prediction of selection response is problematic, in part because individuals are expected to vary in their degree of inbreeding, because the covariance between parent and offspring, on which selection response depends, involves all the dominance components in addition to V_{A} and because allele frequencies and linkage disequilibrium, and hence the components of variance, change more rapidly with inbreeding and selection than with selection alone. Although special cases of selection with inbreeding have been considered (Wright and Cockerham 1985; Kelly 1998), no general analytical treatment accounting for all of these complications is available. Simulations of shortterm selection response, based on our estimates of the variance components for three traits, show that response can be appreciably less than predicted from the breeder’s equation (Equation 1) when traits are subject to the complete dominance model in populations undergoing partial inbreeding. Moreover, even if mating is at random, IBD that accrues because of drift in small populations and that can be increased by selection can significantly affect selection response when autozygous dominance deviations differ from random dominance deviations in their effects on the phenotype.
Results for Model 2, involving inbreeding depression alone (with V_{DI} = V_{DR}), showed reductions in selection response of up to 70% per generation, depending on the mating scheme. Our simulations further showed that when this genetic model applies, shortterm selection in partially inbreeding populations can be well predicted by summing the response expected from the breeder’s equation (Equation 1) and the effect on the mean due to inbreeding depression. We found, moreover, that this approach to predicting selection response closely approximated the average response in our simulations, even with the much larger values of V_{DI} that we found for petal width, as long as Cov(a, d_{I}) was specified as zero. However, when additive effects of alleles are not independent of autozygous dominance deviations (Model 3), striking differences from these predictions arise. Even with the small, negative estimates of Cov(a, d_{I}) we obtained for petal width, response to selection toward larger petals is reduced substantially more than can be simply accounted for by inbreeding depression. Negative Cov(a, d_{I}) implies that the higher the effect of an individual allele, the more extreme tends to be its contribution, in autozygous state, to inbreeding depression, and thus, selection exacerbates inbreeding depression. The simulations show that even the moderate values of Cov(a, d_{I}) we estimated can strongly affect selection response. Given the precision of our estimates, however, we cannot reject the null hypothesis that the true value of this parameter is zero for any trait. If future work consistently fails to demonstrate definitively that Cov(a, d_{I}) contributes to genetic variance in inbreeding populations, then it appears that valid shortterm predictions can be obtained by the composite method above, requiring only estimates of V_{A} and inbreeding depression, both of which can be estimated from genetic designs far simpler than the cumbersome pedigrees required to estimate all the parameters of Model 3.
These dominance effects on selection can be viewed as distinct causes of selection decline in addition to those modeled by Robertson and Hill (1983) for finite populations undergoing selection, drift, and accumulation of negative linkage disequilibrium, leading to reduction in additive genetic variance. Neither of these processes is directly modeled in our simulations, but must be taken into account in longer range predictions of selection response (Weiet al. 1996). Our simulations complement Kelly’s (1998) analytical and simulation results showing reduction in genetic variance and selection response with partial inbreeding. Kelly’s “structured linear model” accounts for the inbreeding coefficient of each individual by partitioning the population into cohorts having different numbers of sequential selfing events in their immediate ancestry. His simulations, therefore, provide insight into selection response in a large population when all inbreeding is due to selfing. When population size is small, or when pollen and seed distribution is spatially limited, consanguineous matings occur in a less predictable manner and result in a broad range of individual inbreeding coefficients. An approach suggested by Cockerham and Weir (1984) to account for variation in inbreeding is inclusion of a further component of variance attributable to variation in individual inbreeding coefficient in a population with a known mean (equilibrium) inbreeding coefficient.
We know of few empirical studies in which effects of nonrandom mating on the structure of genetic variation have been assessed. Cornelius (1988) attempted to estimate the genetic components that may arise under inbreeding in a population of maize, but detected none of them as statistically significant. Similarly, in a study of sheep experimentally inbred in six populations (three representing distinct breeds and three crosses between breeds), Shaw and Woolliams (1999) found little evidence of contributions of genetic components theoretically defined for inbred populations. For live weight of purebred lambs, they did find significant V_{DI} and Cov(a, d_{I}), but the analysis of that particular dataset suffered from bias due to the pooling of several breeds. For the larger dataset of crossbred lambs, where this bias was minimal, substantial inbreeding depression and significant V_{A} and V_{DR} were detected, but estimates of V_{DI} were distinguishable neither from those of V_{DR} nor from zero, even though they were comparable to the estimates of V_{A}. Because the size of this experiment was smaller than the one presented here (e.g., the pooled number of crossbred lambs is 1480), and the design involved a large number of fixed effects accounting for variation in environmental conditions over a span of 20 years, the statistical power of this study may have been insufficient for detection of these novel components.
Taking a distinct approach in a study of Plantago lanceolata, Tonsor and Goodnight (1997) compared estimates of heritability from plants produced by random mating with estimates from progenies arising from matings according to the natural distribution of pollen dispersal. Among nine traits studied, for only reproductive dry weight was the heritability found to be marginally significantly greater under the localized mating scheme. Thus, the limited evidence available differs from our study in suggesting that novel components of genetic variation defined for inbred populations tend to make minor contributions to their phenotypic variance.
Our findings of substantial V_{DI} contributing to variation in three traits expressed under inbreeding suggest that further studies of the components of genetic variance in partially inbreeding populations would be of value. An accumulation of evidence that, apart from V_{DI}, inbred dominance components are negligible could justify appreciable simplification in experimental designs used to study genetic variation under inbreeding. However, it is premature to rule out the importance of the remaining components even in the population of N. menziesii we studied. It remains to be seen whether expression of variation in other characters or under field conditions is subject to gene action involving the remaining components not detected in the present study.
Acknowledgments
Discussions with M. Lynch and D. Houle stimulated our initial work, funded by Pioneer HiBred International; collaboration with J. Woolliams with support from the Biotechnology and Biological Sciences Research Council (BBSRC), U.K. greatly enhanced our progress and understanding. We thank J. Larson for invaluable assistance in many forms and A. Caballero, D. Charlesworth, J.L. Jannink, J. Kelly, R. Miller, A. Montalvo, J. Stone, B. Walsh, N. Waser, and an anonymous reviewer, all of whose comments greatly improved the manuscript. We appreciate support from the National Science Foundation, which funded portions of this project.
Footnotes

Communicating editor: M. K. Uyenoyama
 Received January 26, 1998.
 Accepted September 8, 1998.
 Copyright © 1998 by the Genetics Society of America