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CROP BREEDING, GENETICS & CYTOLOGY

Dominance and Genetic Drift

Predicted Effects of Population Subdivision in a Maize Population

^a Dep. of Biostatistics, Univ. of Alabama at Birmingham, RPHB 327, 1530 3rd Ave. South, Birmingham, AL 35294-0022
^b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011

^* Corresponding author (jode{at}uab.edu).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Many public sector maize recurrent selection programs have been designed based on additive genetic expectations. Populations have been managed as large metapopulations with the assumption that population size must be very large because inbreeding due to finite size causes a linear reduction in genetic variance; we show that in BS13(S)C0 such predictions are inaccurate and discuss some implications. The objective of this study was to predict the effects of subdividing a maize population, BS13(S)C0, into finite subpopulations with previous estimates of genotypic variance-covariance components in the BS13(S)C0 population. Changes in variance among subpopulations, genetic variances within subpopulations, and mean values of subpopulations were predicted. Predicted variance among subpopulations increased approximately linearly with the inbreeding coefficient, in accordance with additive genetic expectations. Additive genetic theory predicts a linear decline in both total and additive genetic variance within subpopulations. Predicted total genetic variance within subpopulations initially increased, then decreased when the inbreeding coefficient was between 0.2 and 0.4 for most traits. Predicted additive genetic variance within subpopulations for grain yield decreased little at inbreeding coefficients <0.5. Predicted additive genetic variance for other traits decreased in approximate accordance with additive genetic expectations. These results provide model-based predictions that inbreeding BS13(S)C0, that is, genetic drift, will not lead to linear reductions in total genetic variance or additive genetic variance as typically expected. Implications of these results for agricultural selection programs are discussed.

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
RECURRENT SELECTION PROGRAMS in public sector maize breeding programs have historically been designed to minimize loss of genetic variability (variance) through avoidance of genetic drift (Hallauer, 1992). The design of such programs is commonly based on additive genetic theory, which states that in a population undergoing inbreeding, genetic variance is reduced linearly in the inbreeding coefficient (Wright, 1951). However, it has been well established theoretically that in the presence of directional dominance or epistasis, inbreeding does not necessarily reduce genetic variance and may actually increase genetic variance (Robertson, 1952; Avery and Hill, 1979; Bryant et al., 1986a,b; Goodnight, 1987, 1988, 1995; Cockerham and Tachida, 1988; Tachida and Cockerham, 1989; Whitlock et al., 1993; Willis and Orr, 1993; Cheverud and Routman, 1995, 1996; Wang et al., 1998). The phenomenon of increasing additive genetic variance with inbreeding, or conversion of nonadditive genetic variance into additive genetic variance, has received immense interest in evolutionary genetics recently, but has been curiously unexplored in plant breeding programs.

Nonadditive genetic effects, directional dominance in particular, has been found to be nearly ubiquitous in maize. Inbreeding depression, an observed outcome of directional dominance, has been measured extensively in maize (Sing et al., 1967; Hallauer and Sears, 1973; Cornelius and Dudley, 1974; Good and Hallauer, 1977; Smith, 1983; Walters et al., 1991; San Vicente and Hallauer, 1993; Benson and Hallauer, 1994). A model proposed by Smith (1983) was used in several studies to measure the effects of inbreeding depression resulting from genetic drift during ongoing selection programs (Tanner and Smith, 1987; Helms et al., 1989a; Eyherabide and Hallauer, 1991; Keeratinijakel and Lamkey, 1993). Hallauer and Miranda Filho (1988) summarized 99 experiments in maize in which genetic dominance variance was measured and found an average estimate of ²_D/²_A for grain yield of 0.61, demonstrating that dominance makes a substantial general contribution to genetic variance for grain yield in maize populations. Given the importance of dominance variance in maize, it is clear that additive genetic expectations will lead to incorrect assumptions concerning changes in genotypic covariance components in populations undergoing inbreeding or genetic drift.

Quantitative models of changes in genotypic covariance components rely on a conceptual model of an effectively infinite metapopulation that becomes subdivided into a large number of isolated, inbred subpopulations (Wright, 1951; Robertson, 1952; Avery and Hill, 1977, 1979; Goodnight, 1987, 1988, 1995; Cockerham and Tachida, 1988; Tachida and Cockerham, 1989; Willis and Orr, 1993; Whitlock et al., 1993; Cheverud and Routman, 1995, 1996; Wang et al., 1998). In plant breeding programs, we are generally interested only in a particular population, such as BS13(S)C0. The conceptual bridge from a model of many subpopulations to a particular population of interest is achieved as described by Wright, (1984): "A single existent population may sometimes be thought of to advantage as one of an infinite number of possible populations that might have been derived from a specified ancestral population." The degree of population subdivision, or inbreeding, of subpopulations and individuals is quantified by Wright's F statistics for hierarchic populations (Wright, 1984). The degree of inbreeding at the subpopulation level is described by F_ST, the correlation between uniting gametes within a subpopulation with respect to the metapopulation (Wright, 1984). Inbreeding of individuals within subpopulations with respect to the subpopulation is quantified by F_IS, the correlation between gametes uniting in individuals with respect to the subpopulation (Wright, 1984). The total inbreeding of individuals with respect to the ancestral metapopulation is given by F_IT (Wright, 1984). Alternative definitions and discussions of the F statistics can be found in common texts (Crow and Kimura, 1970, p. 105; Hartl and Clark, 1997, p. 117).

Changes in genetic variances within and among subpopulations at low values of F_ST (i.e., at the beginning of population subdivision) are affected by dominant gene action in two ways: (i) total genetic variance and additive variance within subpopulations are increasing at low values of F_ST, whereas they decrease linearly in F_ST with additive gene action; and (ii) variance among subpopulations increases much more slowly than expected under an additive genetic model (Robertson, 1952; Willis and Orr, 1993; Wang et al., 1998). Avery and Hill (1979) showed that linkage disequilibrium, generated by finite population size, further inflates dominance variance within subpopulations, which contributes to total genetic variance within subpopulations. Results qualitatively similar to Robertson's (1952) have been shown theoretically for epistatic gene action (Bryant et al., 1986a, b; Goodnight, 1987, 1988, 1995; Cockerham and Tachida, 1988; Tachida and Cockerham, 1989; Whitlock et al., 1993; Cheverud and Routman, 1995, 1996).

Estimates of genotypic covariances demonstrating departures from additivity have been obtained in numerous contexts. Variance among subpopulation means were larger than expected under an additive model in a study of morphometric traits in the housefly, Musca domestica L., (Bryant et al., 1986a) and for viability in Drosophila melanogaster Meigen (García et al., 1994). Increases in additive genetic variance following inbreeding have been demonstrated for morphometric traits in Musca domestica (Bryant et al., 1986b, Bryant and Meffert, 1992), viability in Drosophila melanogaster (López-Fanjul and Villaverde, 1989; García et al., 1994), and viability in Tribolium castaneu (Herbst) (Fernández et al., 1995). While these studies did not have sufficient power to differentiate between dominance and epistasis, inbreeding depression was found in all cases cited for viability (López-Fanjul and Villaverde, 1989; García et al., 1994; Fernández et al., 1995), and for body-size traits in Musca domestica (Bryant et al., 1986b), demonstrating the presence of intralocus dominance deviations and suggesting that segregating rare recessives made an important contribution to observed patterns of genetic variance. Bryant et al. (1986a) showed that differentiation among bottlenecked subpopulations of Musca domestica for body size-associated traits was more consistent with a model of epistasis than a dominance model, but the power of this conclusion was somewhat limited by sample size (Lynch, 1988).

Edwards and Lamkey (2002) have published estimates of genotypic covariance components for inbred and noninbred relatives in the BS13(S)C0 maize population. Regarding BS13(S)C0 as a metapopulation, it is possible to use estimates provided by Edwards and Lamkey (2002) to predict changes in genetic variance structure in subpopulations derived from BS13(S)C0 where subpopulations may include later cycles of recurrent selection, subsamples of the base population, or single full-sib families or S₁ lines. The objective of this study was to use parameter estimates from a quantitative genetic model that accounts for inbreeding and dominance to predict the effects of population subdivision in the BS13(S)C0 maize population. Specifically, we would like to make quantitative predictions of expected changes in (i) additive genetic variance within inbred subpopulations, (ii) total genetic variance within inbred subpopulations, (iii) genotypic covariances between genetic effects in inbred individuals within subpopulations, and (iv) variance among subpopulation means.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Genetic Model
The value g_ij of genotype A_iA_j in a population of individuals in Hardy-Weinberg equilibrium (Fisher, 1918) is

[1]

where g_ij = genetic value of genotype A_iA_j, µ = population mean at panmixia, _i = additive effect of the ith allele, _j = additive effect of the jth allele, and _ij = dominance deviation of genotype A_iA_j.

Harris (1964) and Gillois (1964) extended the classical single-locus genetic model first introduced by Fisher (1918) to include covariances between inbred relatives. The covariance between two individuals X and Y is a linear function of five genotypic covariance components (Gillois, 1964; Harris, 1964):

[2]

where p_i = frequency of ith allele; ²_A = p_i²_i = additive variance; ²_D = p_ip_j²_ij = dominance variance; D₁ = p_iiii = covariance between additive effects and homozygous dominance effects; D^*₂ = p_i²_ii - ² = variance of homozygous dominance deviations; H* = ² = sum of homozygous dominance deviations, squared; and F, F, _XY, _Y, _X, ₊, _·, are probabilities of identity by descent for sets of 2, 3, and 4 alleles (Cockerham, 1971).

Population Structure
Effects of population subdivision were predicted under the assumptions of a monoecious randomly mating population with no overlapping generations, which implies (i) an individual has equal probability of providing male and female gametes; (ii) all possible pairs of individuals have equal probability of mating; and (iii) the probability that an allele is drawn at random from the population is equivalent to its allelic frequency.

It was assumed that population subdivision occurred as follows: (i) Each subpopulation was founded from a finite number of individuals that were randomly sampled from the metapopulation, BS13(S)C0. (ii) Each subpopulation was a monoecious, randomly mating population subject to the mating assumptions outlined for monoecious, randomly mating populations without overlapping generations. (iii) No gene flow occurred between subpopulations. (iv) Nonoverlapping generations of mating occurred within subpopulations until all individuals in a subpopulation contained two identical copies of the same haploid genome (i.e., subpopulations were completely inbred). (v) Effects of mutation are regarded as negligible.

Predicted Changes in Genetic Variances with Population Subdivision
Changes in genetic variances within and among subpopulations were studied by computing predicted variances for all values of F_ST ranging from 0.0 to 1.0 in intervals of 0.05. Predicted genetic variances were then plotted as functions of F_ST for values of F_ST from zero to one to study how predicted genetic variances change with increasing levels of inbreeding within subpopulations. All computations were performed with Mathcad 2001 Professional (Mathsoft Applications, Cambridge, MA). Predictions were computed according to a three-step process that will now be outlined.

Step 1—Compute Expected Multiallele Descent Measures within Subpopulations
In the idealized conditions we assumed for population subdivision, four identity-by-descent probabilities for sets of two, three, and four alleles were sufficient to predict all expected genotypic variances and covariances within and among subpopulations (Chevalet and Gillois, 1976; Cockerham and Weir, 1983). The probability that two alleles are identical by descent, , is equivalent (given the idealized population structure) to Wright's F_ST (Robertson 1952, Cockerham and Weir, 1983). Three- and four-allele identity probabilities can be expressed as functions of with approximations according to Chevalat and Gillois (1976):

[3]

Values of the four descent measures, , , , and , were computed for all values of F_ST from 0.0 to 1.0 in intervals of 0.05.

Step 2—Compute Estimates of Within-Subpopulation Genotypic Covariances
Jiang and Cockerham (1990) have provided landmark theory to predict expected genotypic covariance components within subpopulations as functions of (i) expected multiallele descent measures within subpopulations (computed in Step 1), and (ii) estimates of genotypic covariance components in the metapopulation from which subpopulations were derived (estimates available from Edwards and Lamkey, 2002). Predicted genotypic covariance components within inbred subpopulations are denoted with asterisks in their subscripts, ²_A*, ²_D*, D_1*, D^*_2*, and H^*_*, to distinguish them from metapopulation components, ²_A, ²_D, D₁, D^*₂, and H*. Predicted within-subpopulation genotypic covariance components are functions of the genotypic covariance components in the metapopulation (i.e., the covariance structure of the metapopulation) and of the level of allelic identity-by-descent in the subpopulation (quantified by , , , and ; Jiang and Cockerham, 1990):

and

Estimates of genotypic covariance components were obtained from Edwards and Lamkey (2002) for grain yield, plant height, and grain moisture, and used to predict subpopulation components for those traits in BS13(S)C0. We compared predictions for the three traits in BS13(S)C0 to five two-allele models defined by varying degrees of dominance, d, and recessive allele frequency, q (Table 1) . We defined the degree of dominance, d, as the contrast between the value of a heterozygote and the midpoint of the values of two homozygotes

where x_AA = value of genotype AA, x_Aa = value of genotype Aa, and x_aa = value of genotype aa.

Step 3—Compute Subpopulation Variances and Covariances of Interest
After the complete genotypic variance-covariance structure within subpopulations was computed for a range of values of F_ST, subpopulation variances and covariances of interest were computed for the three traits in BS13(S)C0 and the five two-allele models. Variances and covariances of interest were computed with respect to subpopulations that had undergone one generation of population expansion. Population expansion was defined by the process of obtaining an effectively infinite number of offspring by allowing random mating (subject to assumptions outlined for monoecious random mating) among the finite number of individuals in the subpopulation. The assumed population expansion allows us to avoid inflation of observed variances that arises from sampling of a finite number of individuals, particularly for variance among subpopulation means (V_B). Six genetic variances and covariances of interest in a plant breeding context were examined (Table 2) . Variance among subpopulation means, V_B, was equivalent to the expected covariance between randomly sampled individuals within subpopulations because of the assumption of population expansion (Robertson, 1952; Jiang and Cockerham, 1990). Computation of V_B thus required only expected within-subpopulation probabilities of identity by descent for multiple alleles and metapopulation genotypic covariance components. The remaining five variances in Table 2 were linear functions of within-subpopulation genotypic covariance components described by Jiang and Cockerham (1990). Predicted genetic variance among noninbred individuals (F_IS = 0) within subpopulations, V_W(0), and predicted additive genetic variance within subpopulations, V_AW(0), provided an indication of how genetic variance and heritable genetic variance change within subpopulations as F_ST increases. Total genetic variance among inbred individuals (F_IS = 1) within subpopulations, V_W(1), is the predicted genetic variance among inbred individuals within subpopulations. For inbred individuals within subpopulations (F_IS = 1), it was assumed that after population expansion, a large number of individuals were randomly chosen from the expanded subpopulation and continuously self-pollinated to obtain individuals that were completely inbred with respect to the subpopulation (and metapopulation). The total genetic variance among inbred individuals (F_IS = 1) within subpopulations can be decomposed into heritable [C_GAW(1)] and nonheritable components [C_GDW(1)] by the relation V_W(1) = C_GAW(1) + C_GDW(1), analogously to the decomposition of total genetic variance among noninbred individuals (F_IS = 0) within subpopulations into V_W(0) = V_AW(0) + V_DW(0) (V_DW(0) is the variance of dominance deviations of noninbred individuals within subpopulations). The covariance between inbred genotypic values and their breeding values, C_GAW(1), is a direct indication of the heritability of inbred line performance because it describes the genotypic covariance between performance of the inbred lines per se (genotypic value) and outbred progeny of the inbred line (inbred line breeding value). The covariance between inbred genotypic values and their dominance deviations, C_GDW(1), is a measure of the portion of genetic variance of inbred genotypic value that is not heritable.

	RESULTS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Variance among Subpopulation Means
Predicted variances among subpopulation means for the two-allele models we examined were less than predicted variance among subpopulation means under an additive model in all cases except for overdominance and low recessive allele frequency (d = 2, q = 0.1) at values of F_ST > 0.5 (Fig. 1) . In contrast, predicted variances among subpopulation means in BS13(S)C0 were in close accordance to an additive model for grain yield and grain moisture, while plant height had a larger variance among subpopulation means than expected under an additive model across all values of F_ST (Fig. 2) .

View larger version (26K): [in this window] [in a new window]   Fig. 1. Predicted variance among subpopulation means for five two-allele models characterized by degree of dominance (d) and recessive allele frequency (q). For each model, the variance among subpopulations was divided by the variance among completely inbred subpopulations (VB at FST = 1).

View larger version (23K): [in this window] [in a new window]   Fig. 2. Predicted variance among subpopulation means for grain yield (YLD), grain moisture (MST), and plant height (PHT) in the BS13(S)C0 maize population. For each trait, the variance among subpopulation means was divided by the total variance among completely inbred subpopulations (VB at FST = 1).

View larger version (26K): [in this window] [in a new window]   Fig. 3. Predicted total genetic variance among noninbred individuals [VW(0)] within subpopulations for five two-allele models characterized by degrees of dominance (d) and recessive allele frequency (q). Predicted genetic variances were divided by the total genetic variance among noninbred genotypic values in the metapopulation [VW(0) at FST = 0].

View larger version (27K): [in this window] [in a new window]   Fig. 4. Predicted additive genetic variance within subpopulations [VAW(0)] for five two-allele models characterized by degree of dominance (d) and recessive allele frequency (q). Predicted additive genetic variances were divided by the total genetic variance among noninbred genotypic values in the metapopulation [VW(0) at FST = 0].

View larger version (31K): [in this window] [in a new window]   Fig. 5. Predicted total genetic variance among noninbred individuals within subpopulations [VW(0), labeled VW] and additive genetic variance within subpopulations [VAW(0), labeled VA] for grain yield (YLD), grain moisture (MST), and plant height (PHT) in the BS13(S)C0 maize population. All predicted variances were divided by the total genetic variance among noninbred genotypic values in the metapopulation [VW(0) at FST = 0 and FIS = 0].

and

With these equations, we found patterns of change in V_W(1), C_GAW(1), and C_GDW(1) as F_ST increased from zero to one that were consistent among two allele models and traits [Fig. 6, 7 ; lines for C_GAW(1) denoted by A for breeding value, lines for C_GDW(1) denoted by D for dominance]. Specifically, V_W(1), the predicted genetic variance among inbred individuals (F_IS = 1) within subpopulations, decreases linearly with increasing F_ST independently of the genotypic covariance structure in the metapopulation. Genetic variance among inbred individuals, V_W(1), decreases linearly because it is a linear function of the total genetic variance of inbred genotypic values, 2²_A + 4D₁ + D^*₂, with coefficient 1 - (equivalent to 1 - F_ST). The predicted covariance between inbred genotypic values and inbred breeding values within subpopulations, C_GAW(1), monotonically approaches V_W(1) as F_ST increases (Fig. 6, 7). The predicted covariance between inbred genotypic values and inbred dominance deviations within subpopulations, C_GDW(1), monotonically approaches zero at a rate of V_W(1) - C_GAW(1) (Fig. 6, 7). Hence, as subpopulations become increasingly inbred (increasing F_ST), predicted genetic variance among inbred individuals within subpopulations becomes increasingly heritable because C_GAW(1) (covariance with breeding value) approaches V_W(1) and C_GDW(1) (covariance with dominance deviations) approaches 0. Two allele models (Fig. 6) and traits in BS13(S)C0 (Fig. 7) were consistent in their patterns of change in C_GAW(1) and C_GDW(1) with increasing F_ST, but differed substantially in initial values of C_GAW(1) and C_GDW(1) in the metapopulation. In BS13(S)C0, plant height and grain moisture had initial values of C_GAW(1) close to V_W(1) and initial values of C_GDW(1) close to zero, so there was very little predicted change in contribution of covariances C_GAW(1) and C_GDW(1) to predicted genetic variance among inbred individuals within subpopulations. In contrast, for grain yield, C_GDW(1) was 0.48 compared with C_GAW(1) of 0.21 in the metapopulation (Edwards and Lamkey, 2002). Because of the large value of C_GDW(1) relative to C_GAW(1) in the metapopulation, the relative contribution of C_GDW(1) to V_W(1) was rapidly decreasing at low values of F_ST (Fig. 7) while the relative contribution of C_GAW(1) to V_W(1) was rapidly increasing at low values of F_ST. Hence, even mild inbreeding (population subdivision) was predicted to cause substantial increases in the heritability of inbred genotypic values within subpopulations in BS13(S)C0.

View larger version (27K): [in this window] [in a new window]   Fig. 6. Predicted covariances between inbred genotypic values and inbred breeding values, CGAW(1), (labeled A) and between inbred genotypic values and inbred dominance deviations, CGDW(1), (labeled D) for three two-allele models characterized by degree of dominance (d) and recessive allele frequency (q). Plots are labeled with ordered pairs in parentheses, representing degree of dominance and recessive allele frequency, respectively [e.g., (2, 0.1) represents d = 2 and q = 0.1; allele frequency excluded for d = 0]. All covariances were divided by predicted genetic variance among inbred individuals [VW(1)] at the same value of FST.

View larger version (28K): [in this window] [in a new window]   Fig. 7. Predicted covariances between inbred genotypic values and inbred breeding values, CGAW(1), (labeled A) and between inbred genotypic values and inbred dominance deviations, CGDW(1), (labeled D) for grain yield (YLD), grain moisture (MST), and plant height (PHT) in the BS13(S)C0 maize population. All covariances were divided by predicted genetic variance among inbred individuals [VW(1)] at the same value of FST.

View larger version (22K): [in this window] [in a new window]   Fig. 8. Predicted 95% drift intervals for grain yield in the BS13(S)C0 maize population for values of FST ranging from zero to one. The line µ represents the predicted metapopulation mean, lines µ ± 2 are 95% confidence limits for values of individuals, and lines µ ± 2 are 95% confidence limits for values of subpopulation means.

View larger version (22K): [in this window] [in a new window]   Fig. 9. Predicted 95% drift intervals for grain moisture in the BS13(S)C0 maize population for values of FST ranging from zero to one. The line µ represents the predicted metapopulation mean, lines µ ± 2 are 95% confidence limits for values of individuals, and lines µ ± 2 are 95% confidence limits for values of subpopulation mean.

	DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Robertson (1952) described predicted changes in genetic variance following population subdivision with dominant gene action as compared with predictions with completely additive gene action. We have applied genotypic covariance theory to make the same predictions in a maize population, BS13(S)C0. Defining an appropriate additive expectation for traits measured in a real population is a major challenge for making this comparison because allele frequencies and allelic effects are unknown in the population. For example, under an additive model, additive genetic variance is equal to total genetic variance in the base population. In an observed population in which total genetic variance is greater than additive genetic variance [such as BS13(S)C0], it was unclear to us which was more appropriate as an additive expectation, total genetic variance (all of which is expected to be additive under an additive model) or additive genetic variance [which is expected to be equal to total genetic variance under an additive model, but clearly was not in BS13(S)C0]. We conjecture that the most appropriate comparison could be determined only if we could answer the following hypothetical question: If all dominance coefficients (d) of alleles in BS13(S)C0 could be changed to zero (without affecting allelic frequencies or homozygote contrasts), what would the genetic variance be? We chose to use the total genetic variance for the comparison in Fig. 4 and 5. The appropriate additive expectation and scale on which to examine all genetic variances and covariances we examined is open to debate. We chose the scales that appeared most appropriate for clear presentation and for making specific points. However, future application of this theory would benefit from clearly identifying the most appropriate null expectations.

The full implications of Robertson's (1952) description of the nonlinear relationship between genetic variance and inbreeding have not been well recognized among plant breeders. It is a common assumption that inbreeding associated with routine maintenance of populations or recombination of finite numbers of lines during selection programs will lead to linear reduction in genetic variance. However, our predictions showed that population subdivision (formation of inbred subpopulations) in BS13(S)C0 is expected to increase total genetic variance for grain yield and maintain additive genetic variance near its initial level up to an inbreeding coefficient of >0.5. Several studies of Iowa Stiff Stalk Synthetic populations have found very little change in genetic variance after several cycles of recurrent selection (Helms et al., 1989b; Lamkey, 1992; Stucker and Hallauer, 1992; Holthaus and Lamkey, 1995), even though it has been noted that reductions in variance were expected (Holthaus and Lamkey, 1995). Although our predictions apply strictly only to inbreeding without selection, there is still noteworthy qualitative correspondence between our predictions and observed results in Iowa Stiff Stalk Synthetic populations. Namely, heritable genetic variance for grain yield is not linearly exhausted in either case. Our predictions with respect to changes in C_GAW(1) and C_GDW(1) further suggest that genetic drift in BS13(S)C0 will gradually improve the genotypic covariance structure of the subpopulations as F_ST increases. This is because the covariance between inbred genotypic values and breeding values, C_GAW(1), gradually contributes more to V_W(1) while the covariance between inbred genotypic values and dominance deviations, C_GDW(1), gradually approaches zero with increasing F_ST. The increasing covariance between inbred genotypic value and breeding value, C_GAW(1), implies that selection on inbred individuals will increasingly be more effective at improving outbred progeny performance as F_ST increases. In direct correspondence with these predictions, Lamkey (1992) reviewed progress from S₂–progeny recurrent selection in BS13(S)C0 and found an observed selection response pattern that may be consistent with such a scenario; in early cycles of selection, no response or negative response was observed, followed by improved response in later cycles (Lamkey, 1992). We cannot say with any certainty that our predictions established that this should have been the case, especially given that there is only a single replicate of the selection program, but the qualitative correspondence between our predictions and observed patterns supports further development of this theory for application to plant breeding. Our results have two very important implications for selection on traits controlled by nonadditive gene action: (i) narrow-based populations may be expected to have comparable or even greater genetic variance and or improved genotypic covariance properties than the metapopulation from which they were derived, and (ii) ample genetic variance can be maintained during inbreeding. Even though these results do not extend quantitatively to short-term selection programs because we have not accounted for the combined effects of inbreeding and selection, the results do draw attention to the fact that we cannot assume genetic variance will decline linearly with inbreeding.

At the same time that our predictions establish maintenance or even increased genetic variance within inbred subpopulations, our predictions showed that the increase in variance among subpopulations is approximately linear in the inbreeding coefficient. In our examination of genetic drift intervals (Fig. 8, 9) that showed the predicted range of randomly sampled subpopulations and individuals as a function of the inbreeding coefficient, F_ST, the variance among subpopulations makes a significant contribution to total genetic variation even at small inbreeding coefficients. From a viewpoint of selection, these drift intervals are graphic depictions of the importance of selection among subpopulations. A topic that needs much greater formal attention in plant breeding is selection among and within subpopulations; that is, selection in subdivided populations. Our results show that intentional subdivision of BS13(S)C0 into discrete subpopulations could result in potentially greater additive genetic variance within subpopulations, complemented by variability among subpopulations that could be exploited by selection both within and among subpopulations.

Another issue of concern to breeders is whether increased genetic diversity is needed to increase genetic variance for selection response. Diversity, in a mathematical sense, is a function of allelic frequencies, whereas inbreeding is a function of allelic identity by descent. Our results predict changes in genetic variance when the level of identity by descent increases. Increased identity by descent in a population necessarily reduces diversity within the population. Hence, forward predictions such as those we have made can be applied to the question of what happens to genetic variance if diversity is reduced from its present level by inbreeding. However, the theory doesn't apply in the backward direction; that is, we cannot predict how genetic variance is expected to change if a metapopulation becomes more diverse; by definition, the metapopulation is noninbred (F_ST = 0) and hence cannot become less inbred. On the other hand, if a particular population (with potentially small genetic variance) is regarded as a subpopulation of an ancestral metapopulation, comparison of the subpopulation to the ancestral metapopulation can be made. If anything is known about the ancestral metapopulation, the theory can be applied in the forward direction to model the predicted change in genetic variance that was expected to occur in obtaining the contemporary subpopulation. This cannot, however, provide any predictions about the effect of outcrossing the contemporary subpopulation to unrelated individuals or subpopulations. Hence, the value of this theory is in predicting how increased inbreeding will change expected genetic variance, but it is much less insightful on the issue of increasing the diversity of a population.

Genetic diversity is a requirement for genetic variance and response to selection. We have utilized quantitative genetic theory to predict that random inbreeding, which reduces diversity, may have desirable quantitative genetic effects. If random inbreeding has desirable effects, we can further conclude that a population may be too diverse. It follows that too much diversity may be just as damaging to selection response as not enough. Hence, from a quantitative genetics point of view, we have shown that increased diversity by itself cannot be expected to increase genetic variability in a manner that will lead to an increased rate of genetic improvement.

	ACKNOWLEDGMENTS

 
We thank Paul Cornelius and Charles Goodnight for reviewing an earlier version of this manuscript and providing us with helpful comments.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Joint contribution from the USDA-ARS Field Crops Res. Unit and journal paper of the Iowa Agriculture and Home Economics Exp. Stn., Ames, IA; Project no. 3755, and supported by the Hatch Act and State of Iowa funds.

Received for publication October 15, 2002.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 

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