Abstract
A method to predict longterm genetic contributions of ancestors to future generations is studied in detail for a population with overlapping generations under mass or sib index selection. An existing method provides insight into the mechanisms determining the flow of genes through selected populations, and takes account of selection by modeling the longterm genetic contribution as a linear regression on breeding value. Total genetic contributions of age classes are modeled using a modified gene flow approach and longterm predictions are obtained assuming equilibrium genetic parameters. Generation interval was defined as the time in which genetic contributions sum to unity, which is equal to the turnover time of genes. Accurate predictions of longterm genetic contributions of individual animals, as well as total contributions of age classes were obtained. Due to selection, offspring of young parents had an aboveaverage breeding value. Longterm genetic contributions of youngest age classes were therefore higher than expected from the age class distribution of parents, and generation interval was shorter than the average age of parents at birth of their offspring. Due to an increased selective advantage of offspring of young parents, generation interval decreased with increasing heritability and selection intensity. The method was compared to conventional gene flow and showed more accurate predictions of longterm genetic contributions.
MOST natural and artificial populations have overlapping generations. When generations overlap, the generation interval differs from the cohort interval. In quantitative genetics, generation intervals are generally defined as the average age of parents at birth of their offspring. In this definition, generation interval is based on the contributions of parental age classes to newborn offspring; i.e., the average age of parents is calculated as the sum of ages at birth of offspring weighted by the contribution of each age class to newborn offspring. This approach is adopted in the wellknown gene flow procedure (Hill 1974). However, if selective advantage (e.g., breeding value) is partly inherited, selection in subsequent generations may affect the genetic contribution of parental age classes to future generations. Thus there may be a difference between generation interval based on contributions to newborn offspring, and generation interval based on contributions to future generations. It has been suggested, therefore, to calculate generation intervals on the basis of selected offspring only (Bichardet al. 1973). However, contributions of ancestors to future generations may still deviate from contributions to selected offspring.
Recently, Woolliams et al. (1999) found significant differences between generation interval calculated as the average age of parents at the time of birth of a cohort of offspring and generation interval based on the concept of longterm genetic contributions. The latter concept was first introduced by James and McBride (1958) and developed further for the prediction of inbreeding by Wray and Thompson (1990) and Woolliams et al. (1993). Predictions for more advanced selection systems, however, resulted in complicated expressions (Wrayet al. 1994) due to the recursive nature of the prediction procedure. Working on the infinitesimal model (Fisher 1918), Woolliams et al. (1999) obtained a simple closedform approximation for the prediction of longterm genetic contributions by considering Bulmer’s (1971) equilibrium genetic parameters, which makes a recursive algorithm redundant. The method of Woolliams et al. (1999) covers both discrete and overlapping generations and is applicable to mass selection, index selection, and best linear unbiased prediction selection.
The aim of the current article is twofold. First, two methods of Woolliams et al. (1999) for the prediction of longterm genetic contributions in populations with overlapping generations are studied in detail. They illustrate mechanisms that determine the development of pedigree, the contribution of different categories to the genetic makeup of the population in the long term, and the turnover time of genes. The dependency of longterm genetic contributions and generation intervals on selective advantage is illustrated in populations with overlapping generations under mass or sib index selection, assuming the infinitesimal model (Fisher 1918).
Second, predictions based on the methods of Woolliams et al. (1999) are compared to predictions of longterm genetic contributions and generation intervals based on contributions to unselected newborn offspring, as obtained from conventional gene flow (Hill 1974). Both methods are compared to results obtained from simulated data. Accurate predictions of longterm genetic contributions are an important step toward the prediction of rates of inbreeding in selected populations (Woolliams 1998). The current article focuses on the prediction of genetic contributions and generation intervals; the prediction of rates of inbreeding is in a subsequent article. To show the power of theory of Woolliams et al. (1999), predictions of genetic gain based on longterm genetic contributions are also presented, but this is not the main item, as accurate predictions of genetic gain are already well established (e.g., Villanuevaet al. 1993).
METHODS
Here we first describe the population structure that was used. Subsequently we describe the concept of longterm genetic contributions and the method of Woolliams et al. (1999) for the prediction of longterm genetic contributions in populations with overlapping generations, followed by a description of the relationship between generation interval and genetic contributions. Finally, we describe iterative deterministic and stochastic methods to estimate parameters that are needed to predict longterm genetic contributions and related parameters.
Population model: This section describes the genetic model, population structure, and selection strategy for which predictions of genetic contributions were made. The trait considered was assumed to be determined by the infinitesimal model (Fisher 1918). Phenotypic values (P) were the sum of additive genetic values (A, breeding values) and environmental values (E), i.e., P = A + E. The population consisted of overlapping generations, and selection was based upon a sib index for a single trait. With parents up to a maximum of c_{max} of age there are 2c_{max} categories, one for each sex and age of parent. Categories are indexed by k or by l, so k = 1... c_{max} are males, and k = c_{max} + 1... 2c_{max} are females. Let age(k) denote the age of category k [so age(1) = 1 = age(c_{max} + 1)] and let n_{k} be the number of parents selected from category k. The total number of male and female parents equalled N_{m} =
Basic approach for prediction of longterm genetic contributions: This section introduces the concept of longterm genetic contributions. The longterm genetic contribution (r_{i}) of ancestor i in cohort t_{1} is defined as the proportion of genes present in all individuals in cohort t_{2} deriving by descent from i, where (t_{2}  t_{1}) ∞ → (Woolliamset al. 1993). In other words, the longterm genetic contribution of an ancestor is the ultimate proportional contribution of the ancestor to generations in the distant future. After several generations, genetic contributions of ancestors stabilize (longterm contributions are reached) and become equal for all individuals in that and subsequent generations of descendants, but values differ between ancestors (Wray and Thompson 1990).
In the remainder of the current article, longterm genetic contributions of ancestors are referred to as “genetic contributions,” unless explicitly stated otherwise. Applying the approach adopted by Woolliams et al. (1999), contributions of ancestors are predicted by conditioning on the selective advantage of those ancestors. Since sib indices are used here, the selective advantage is equal to the true breeding value of the ancestor [the only parental effect affecting selection of the offspring is the breeding value of the parent (Wrayet al. 1994)]. For an individual in category l, E(r_{i}_{(}_{l}_{)}A_{i}_{(}_{l}_{)}) ≈ u_{i}_{(}_{l}_{)} = α_{l} + β_{l} (A_{i}_{(}_{l}_{)}  Ā_{l}), where α_{l} is the expected contribution of an average parent in category l, β_{l} is the (A_{i}_{(}_{l}_{)}), and Ā_{l} is the mean breeding value of selected contemporaries of i in category l. For discrete generations, the complication of categories can be ignored and α is obtained directly from the number of parents: α = (2N_{x})^{1}, (x = m, f; Wray and Thompson 1990). For both discrete and overlapping generations, solutions for β can be obtained from two regression models (Woolliams 1998; Woolliamset al. 1999): first, the regression of the number of selected offspring on the breeding value of the parent (λ), and second, the regression of the breeding value of selected offspring on the breeding value of the parent (π). Both λ and π can be computed on the basis of known parameters; a derivation is in appendix a. Under equilibrium genetic parameters (Bulmer 1971), regression coefficients (α, β, λ, π) are equal for the parental and offspring generation, allowing for the following closed form expression to compute β instead of a recursive algorithm (Woolliams 1998):
Prediction of expected longterm genetic contributions in populations with overlapping generations: This section describes the approach of Woolliams et al. (1999) to predict longterm genetic contributions for populations with overlapping generations. For ancestor i in category l, the expected longterm genetic contribution was predicted from u_{i}_{(}_{l}_{)} = α_{l} + β_{l}(A_{i}_{(}_{l}_{)}  Ā_{l}). Predictions of genetic contributions are obtained using a modified gene flow matrix (G) of dimension 2c_{max} × 2c_{max}, which identifies the origin of genes of selected instead of newborn offspring. If the conventional gene flow matrix (Hill 1974) is denoted by G_{0}, elements
Solutions for α and β were obtained from the basic equations (Woolliamset al. 1999),
Improved modified gene flow: A firstorder correction to Equation 1 was derived by taking account of differences among average breeding values of parental subgroups present in the selected offspring (Woolliamset al. 1999). When newborn offspring are grouped according to the category of parents, mean breeding values may differ between those groups. Selection then favors offspring descending from parental categories with a higher breeding value, increasing the genetic contribution of these categories. This phenomenon is fully accounted for by the modified gene flow matrix G, identifying the origin of selected offspring. However, after selection, mean breeding values of selected offspring may still differ between parental category subgroups. This affects the contribution of categories, which was ignored in Equation 1. Improved prediction equations were obtained by conditioning on the parental category in Equation 1 (Woolliamset al. 1999),
Generation interval: Generation interval (L) is defined as the turnover time of genes, i.e., the average time interval between two meioses in which an average gene in the population is involved. This interval is equal to the time in which longterm genetic contributions sum to unity, i.e., the genetic contribution summed over all ancestors entering the population over a time period of L years equals unity: Σ_{L}u_{i} = 1. The generation interval (in years) is therefore equal to the reciprocal of the total longterm genetic contribution per year, i.e., summed over all ancestors per year. In u_{i}_{(}_{l}_{)} = α_{l} + β_{l}(A_{i}_{(}_{l}_{)}  Ā_{l}), the term β(A_{i}_{(}_{k}_{)}  Ā_{k}) is zero on average, the sum of genetic contributions is therefore equal to
Deterministic prediction procedure: Elements of Equations 3, 4, 5, 6, 7 were obtained using an iterative procedure, which is described in this section. The iterative procedure is needed because elements (e.g., variances, genetic gain, and genetic contributions) are mutually dependent and Bulmer’s (1971) equilibrium parameters can only be reached by iteration. [Predictions can also be obtained using base generation parameters, but more accurate predictions are obtained using equilibrium parameters (Woolliamset al. 1999).] Predictions of genetic contributions shown in results are based on Bulmer’s (1971) equilibrium parameters. A numerical example is in appendix c.
Phenotypic variance in year t was the sum of additive genetic variance and environmental variance,
Betweensire family additive genetic variance was calculated from
To calculate elements of the modified gene flow matrix, we need to find how the predefined selected proportion of individuals in category k (p_{k}) is distributed across the parental age subgroups. The kth row of G, therefore, was obtained by finding a common index truncation point for all parental subgroups represented among the selection candidates in category k (separate for male and female parents). The solution for the common truncation point has to satisfy the equations (omitting subscript t for simplicity)
Elements of D are d_{kl} = E[(A_{(}_{i}_{)}_{k}  Ā_{k}) given i has category l parent] and were calculated as (omitting subscript t for simplicity)
Elements of Π were calculated as
As described in the section on prediction of longterm genetic contributions, α can be obtained as a right eigenvector from Equation 3 for the “modified gene flow” and from Equation 6 for the “improved modified gene flow.” In general, eigenvectors can be scaled, i.e., if x is an eigenvector of matrix A with an eigenvalue γ, then nx will also be an eigenvector of A with the same eigenvalue γ. With the same eigenvalue, therefore, different eigenvectors can be obtained from Equations 3 or 6, and an additional constraint has to be imposed. Because contributions have to sum to unity per generation, the eigenvector was scaled accordingly. Therefore, first generation interval was calculated as the average age at birth of offspring weighted by the longterm genetic contribution of the categories (n_{k}α_{k}):
Using E(ΔG) =
Stochastic simulation: To draw inferences on the accuracy of predicted genetic contributions, the breeding scheme described in the Population model section was lected base population of the appropriate family structure was generated. Breeding values of base population animals were taken from
For the calculation of genetic contributions, an ancestor cohort t_{1} was chosen when Bulmer’s (1971) equilibrium genetic parameters were reached. Repeated cycles of selection and random mating were performed until genetic contributions were converged and a descendant cohort t_{2} was chosen. Convergence time of genetic contributions (t_{2}  t_{1}) was approximately equal to 7c_{max}. The longterm genetic contribution of ancestor i in category l in cohort t_{1} to individuals in cohort t_{2} was obtained by summing contributions via all pedigree paths leading from i to individuals in
Genetic contributions were analyzed using the model r_{i}_{(}_{l}_{)} = α_{l} + β_{l}(A_{i}_{(}_{l}_{)}  Ā_{l}) + e_{i}_{(}_{l}_{)}. α was estimated as
RESULTS
In this section, a comparison is made between results from conventional gene flow (Method C; Hill 1974), simple modified gene flow (Method M, Equations 3 and 4), and improved modified gene flow (Method P, Equations 4 and 6), for mass and sibindex selection.
Mass selection
Accuracy of α: Table 1 shows longterm genetic contributions of categories (n_{k}α_{k}) obtained from conventional gene flow (Hill 1974) from Method C, Method M, Method P, and from simulation, for a population with three age classes, with 20 sires in age class 1, 10 dams in age class 1, and 30 dams in age class 3, i.e., N = diag{20,0,0,10,0,30}. This scheme, with a high proportion of dams selected from the oldest age class, was chosen because it clearly illustrates the effect of selective advantage on contributions of categories.
Results from Method C are independent of heritability
Comparing Methods M and P to simulation results shows that the firstorder correction improves the accuracy of the predicted longterm genetic contributions. In Equation 3, differences between selective advantage of selected offspring from different parental categories (d_{kl}) are ignored, resulting in underprediction of contributions of young categories and in overprediction of contributions of older categories (except for
Accuracy of β: Table 2 shows the regression coefficients of contributions on breeding values (β), from Method M, Method P, and from simulation, for N = diag{20,0,0,10,0,30}. Most predictions from Method P are within three times the standard error of simulation results, and the trends in predictions agree well with simulation results. Method P was slightly more accurate than Method M, particularly when modeling the differences between 1 and 3yrold females, i.e., β_{4} and β_{6}. In Method C, the effect of selective advantage is not modeled, i.e., β is implicitly zero.
Accuracy of genetic gain and generation interval: Table 3 shows genetic gain per year and generation interval from Method C, Method M, Method P, and from simulation, for N = diag{20,0,0,10,0,30}. Generation interval was calculated from Equation 7. For Method C, generation interval from Equation 7 is identical to the average age of parents when their progeny are born and is obtained from G_{0}. Generation intervals based on the average age of parents of selected offspring, as suggested by Bichard et al. (1973), are obtained from G (see example in appendix c) and are also in Table 3. Method C does not account for the effect of selection on genetic contributions and therefore results in higher generation intervals than simulation. For the scheme in Table 3, most dams are selected from the oldest category, which increases differences between Method C and Method P. Even when the numbers of females selected were exchanged, however, i.e., N = diag{20,0,0,30,0,10}, there were differences between generation intervals from Method C and Method P (see Figure 2). Method M showed systematic overprediction of generation intervals, which agrees with the overprediction of contributions of older categories (see Table 1). Predicted generation intervals based on the average age of parents of selected offspring, i.e., from G rather than G_{0}, were very close to generation intervals from Method M. Generation intervals from Method P were close to simulation results, only showing minor underprediction for high heritabilities.
For this particular scheme, genetic gain from Method C was more accurate than gain from Method P. However, this was not a general result; e.g., for N = diag {20,0,0,30,0,10} (results not shown) it was the other way around. In general, both methods showed similar accuracies for predicting genetic gain.
Effect of heritability and selection intensity on α: The effect of heritability and selection intensity on average genetic contributions of categories (n_{k}α_{k}) was studied using Method P. Figure 1 shows the predicted longterm genetic contribution of 1yrold females as a proportion of the total contribution of females (n_{4}α_{4}/ (n_{4}α_{4} + n_{6}α_{6})), for two different breeding schemes and for two selection intensities. The breeding schemes were S_{1}: N = diag{20,0,0,30,0,10} and S_{2}: N = diag{20,0,0,10,0,30}. Selection intensity was varied by varying the number of tested offspring per dam, i.e., n_{o} was 4 or 20. To illustrate the relation between genetic contributions and generation interval, Figure 2 shows the corresponding generation interval. In S_{1} and S_{2,} males are selected from a single age, and L is directly related to n_{4}α_{4}/(n_{4}α_{4} + n_{6}α_{6}). Results from Method C are identical to results for h^{2} = 0.
Figure 1 clearly shows an increased contribution of 1yrold females when heritability increases, which is due to an increased selective advantage of offspring descending from 1yrold dams when
The relative longterm genetic contribution of 1yrold females also increased with n_{o} (see Figure 1), i.e., with selection intensity. This is partly due to increased genetic gain resulting in an increased selective advantage of newborn offspring of 1yrold dams, in the same way as when
Effect of selection intensity on β: Figure 3 shows the relation between selection intensity and β for a scheme with N = diag{20,0,0,20,0,20} using Method P. Selection intensity is equal for all categories in this scheme, and was varied by varying the number of tested offspring per dam from n_{o} = 2 (i = 0.798) to n_{o} = 40 (i = 2.336).
Figure 3 shows an increase in β_{1} and β_{4} with increasing selection intensity. On average, β is expected to increase with selection intensity because the regression of selected number of offspring on breeding value (λ) increases with selection intensity (see appendix a) and β is positively related to λ (see Equation 2), explaining the trend for β_{1} and β_{4}. For β_{6} the increase with selection intensity is counteracted by the reduced total contribution of 3yrold dams (see Figure 1). For other heritabilities (results not shown) the relation between β and selection intensity was similar.
Effect of heritability on β: Figure 4 shows the relation between β and heritability using Method P. For
It is a general conclusion for mass selection, therefore, that β of younger categories will increase with
The regression coefficient for 1yrold males (β_{1}) shows only minor variation with
Selection on a sib index
Longterm genetic contributions of categories (n_{k}α_{k}) are mainly dependent on the modified gene flow matrix. For a sib index, G is determined by genetic gain and selected proportions, in the same way as for mass selection. The main differences between sib index and mass selection are, therefore, in the regressions λ and π, resulting in different predictions for β. Results for a sib index, therefore, focus on β, though α will also differ from results for mass selection.
Accuracy of β: Predictions for a sib index are compared to simulation results for two opposite schemes: a scheme with positive weight on family information and a scheme with negative weight on family information. The weights used are different from the classical selection index weights (Hazel 1943), but as shown by Villanueva and Woolliams (1997), optimum index weights for intermediate and longterm responses are generally different from classical index weights.
For positive weight on family information, Table 4 shows β from Method P and from simulation for N = diag{20,0,0,10,0,30}, b_{1} = 1, b_{2} = 1.5, and b_{3} = 2 (i.e., I = P + ½P¯_{FS} + ½P¯_{HS}). In Table 4, Method P shows the same trend as simulation results, but tends to slightly overestimate regression coefficients for 1yrold parents (β_{1} and β_{4}). Predictions of α (results not shown) were close to simulation results and showed similar trends as for mass selection.
For negative weight on family information, Table 5 shows β from Method P and from simulation, for N = diag{20,0,0,10,0,30}, b_{1} = 1, b_{2} = 0.5, and b_{3} = 0 (i.e., I = P  ½P¯_{FS}  ½P¯_{HS}). In Table 5, Method P shows the same trend as simulation results and is accurate. Predictions for α (results not shown) were very accurate, i.e., within ±3 SE with 500 replicates in the simulation.
Effect of index weights on β: Figure 5 shows the effect of a varying emphasis on family information in the selection index on the regression coefficients of longterm genetic contributions on breeding values, for 1yrold male parents (β_{1}), from Method P (lines), and from simulation (markers) for N = diag{20,0,0,20,0,20}. For this scheme, β_{1} gives a good impression of the average level of β, because males are selected from a single category, i.e., there is no competition between categories going on. In Figure 5, the index weights vary from b_{1} = 1, b_{2} = b_{3} = 0, representing complete withinfamily selection, to b_{1} = 1, b_{2} = 2, b_{3} = 2, which is identical to I = P + P¯_{FS}.
For withinfamily selection, β equals zero because offspring are selected on their Mendelian sampling term, which by definition is independent of the parental breeding value. Therefore, selective advantage is not inherited and results (both α and β) are identical to results from Method C.
When index weights on family information increased, β_{1} increased because selection of offspring is increasingly affected by the parental breeding value. Similar relations between the average level of β and weight given to family information were found for other distributions of parents across categories (including schemes with competition between categories).
When weight on family information increases, selection tends to selection of families instead of individuals, whereas λ is derived assuming a continuous linear change. Accuracy of predictions decreased, therefore, when weight given to family information became high, which is shown by the increased difference between lines and markers in Figure 5.
DISCUSSION
This article has studied in detail two methods proposed by Woolliams et al. (1999) for the prediction of longterm genetic contributions of individuals in selected populations with overlapping generations. The methods enable accurate predictions of longterm genetic contributions of individual animals and of categories using a simple linear model. Predictions of genetic contributions within categories were first shown by Woolliams et al. (1999) but never studied in detail. Genetic contributions were predicted conditional on breeding value and category of the ancestor by using a modified gene flow approach. The method accounts for the inheritance of selective advantage both between and within categories, resulting in more accurate predictions of genetic contributions and generation intervals than methods based on contributions to newborn offspring in the next cohort. Some trends in the prediction errors remained (e.g., Table 1, Figure 5), but this is merely a matter of improving the relevant regression equations; they do not undermine the basic ideas underlying the theory. Conventional methods ignore the effect of selection on genetic contributions and therefore underestimate contributions of younger categories and overestimate generation interval. Thus, improved methods were necessary.
Accurate predictions of longterm genetic contributions for overlapping generation schemes facilitate deterministic prediction of rates of inbreeding for these schemes (Woolliams 1998) and consequently enable a computationally feasible optimization of breeding schemes with restricted inbreeding. The modified gene flow approach enables prediction of individual longterm genetic contributions [by including β_{k}(A_{i}_{(}_{k}_{)}  Ā_{k}) in the model for expected contributions], whereas conventional gene flow only enables prediction of average genetic contributions (i.e., assuming β = 0). For the prediction of rates of inbreeding it is crucial to account for the effect of selection between individuals (Wrayet al. 1990), and conventional gene flow is therefore not suitable for prediction of rates of inbreeding.
In the present study, generation interval was defined as L = 1/Rn_{k}α_{k}, i.e., the generation interval is the time in which longterm genetic contributions sum to unity. Intuitively, this is a sensible definition: One generation is the time in which the genes are turned over once. The definition of generation interval as the time in which contributions sum to unity is general and is also applicable to generation intervals based on newborn progeny or on selected progeny. For example, generation interval based on newborn progeny, i.e., the average age of parents when progeny are born, can also be calculated as L_{0} = 1/Rα_{0}, where α_{0} are contributions obtained from conventional gene flow. Generation interval based on contributions to selected offspring only (L_{1}), i.e., the average age of parents of selected offspring, can be obtained from the modified gene flow matrix G (see appendix c) and was close to results from simple modified gene flow. When genetic gain is made and selective advantage is inherited, generation interval based on longterm genetic contributions is shorter than both L_{0} and L_{1}, because selective advantage is partly passed on to more distant offspring.
Whereas L_{0} and L_{1} are based on contributions at a specific time point, i.e., before and immediately after selection of the offspring, L is based on converged, i.e., asymptotic longterm genetic contributions of parental categories, which are an invariable property of a population once contributions have converged. Therefore, the definition of generation interval based on longterm genetic contributions is equal to the turnover time of genes, i.e., it is the average time interval between two meioses, and it is of a more genetical and less operational nature than L_{0} and L_{1}.
In the present study, results are only presented for situations where the selection index of an animal was constant across ages. In practice, animals in different categories will often have different amounts of information, affecting the variance of the selection index. This will mainly affect the G matrix, but is easily accounted for by using index variances specific to categories in the equations presented in methods. The problem is more complex for the prediction of rates of inbreeding, because in that case the lifetime genetic contribution of an ancestor, i.e., its contribution summed over all categories it belonged to over its entire life, is relevant, which requires the probability that the same animal was selected in multiple categories.
Large differences were found between predicted genetic contributions from conventional and from modified gene flow in the present study. These differences were partly caused by the distribution of parents across categories; i.e., in Tables 1 and 3 the majority of the dams were selected from the oldest category. When animals are selected by truncation across categories, differences in generation interval between the two methods will be much smaller. For example, for
In the present article, the withinfamily variance was assumed to be constant over time, which is not strictly true when inbreeding is accumulating. However, genetic contributions are mainly determined in the first few generations, where the inbreeding effects on descendants are still small. Longterm genetic contributions are therefore hardly affected by a reduction of variance due to inbreeding. Furthermore, ignoring the effect of inbreeding on the variance allows for the assumption of Bulmer’s (1971) equilibrium variances (assuming the infinitesimal model), which greatly simplifies prediction equations for longterm genetic contributions (Woolliamset al. 1999). For extremely small populations, e.g., with fewer than five parents per sex, it may become important to account for the effect of inbreeding when predicting longterm genetic contributions.
The number of parents is no guarantee for the genetic constitution of populations in the long term, because selective advantage of parents is inherited by offspring. This is a point of concern for conservation genetics where genetic improvement is also being sought. Simply increasing the number of parents may not safeguard the genetic diversity of a population when offspring of the additional parents have a low chance of being selected. The inheritance of selective advantage is crucial in the prediction of longterm genetic contributions, and thus for the prediction of inbreeding (Wray and Thompson 1990). Recently, Nomura (1997) studied inbreeding in open nucleus breeding systems with discrete generations, assuming that genetic contributions of parental groups (nucleus and commercial animals) to progeny remain unchanged after selection. As recognized by Nomura (1997), this is a critical assumption, and especially in populations with overlapping generations it is likely to be strongly violated.
Asymptotically, response from conventional gene flow is equal to response obtained using the wellknown result of Rendel and Robertson (1950; Hill 1974). When gain obtained from conventional and modified gene flow was compared to simulation results, predictions from both methods showed similar accuracy. For the prediction of genetic gain, the ratio of selection differential over generation interval is crucial, rather than the definitions of selection differential and generation intervals separately. When generation interval is defined as the average age of parents of all offspring, and selection differential is defined as the deviation of selected parents from the overall mean, valid predictions for genetic gain are obtained (James 1977). Conventional gene flow, therefore, is a valid method for predicting genetic gain. The relevance of the current theory lies in predicting the development of pedigree, i.e., of the origin and turnover rate of genes, and in predicting rates of inbreeding; it does not primarily predict genetic gain.
APPENDIX A
Derivation of Λ: Elements λ_{kl} are obtained as
Derivation of Π: Elements π_{kl} are obtained as π_{kl} = Cov(A_{i}_{(}_{l}_{)},A_{j}_{(}_{k}_{)}^{*}/Var(A_{i}_{(}_{l}_{)})^{*}, where ^{*} denotes (co)variances after selection of the offspring. Using Cochran (1951),
APPENDIX B
Derivation of DG: Genetic gain is obtained from
APPENDIX C
Example for mass selection: Consider a mass selection scheme (b_{1} = b_{2} = b_{3} = 1) with three age classes, N = diag{20,0,0,10,0,30},
Contributions and generation intervals from conventional gene flow are n_{1}α_{1} = 0.2857, n_{4}α_{4} = 0.0714, n_{6}α_{6} = 0.2143, L = 1/Rα = 1.75. Equations in the deterministic prediction procedure section were iterated until equilibrium variances were reached, resulting in σ^{2}_{A}_{(m)} = 0.0630,
Categories without parents are not relevant, and have zeroes. G identifies the origin of selected offspring; e.g., g_{14} = 0.205 means that a proportion of 2 × 0.205 = 0.410 of the selected 1yrold males (category 1) descends from 1yrold dams (category 4), i.e., were born when their dam was 1 yr old. From G, the generation interval based on selected offspring equals L_{1} = ½ {0.5 + 0.205 + 3 × 0.295} + ½ {10/40 × (0.5 + 0.223 + 3 × 0.277) + 30/40 × (0.5 + 0.193 + 3 × 0.307)} = 1.595. D represents the breeding value of selected subgroups as deviation from the total selected group, e.g., d_{46} = 0.138 means that 1yrold selected females descending from 3yrold dams have an average breeding value of 0.138 units below the average of all selected 1yrold females.
Solutions from Method M were (Nα)^{T} = (0.312, 0, 0, 0.124, 0, 0.188), β^{T} = (0.0201, 0, 0, 0.0161, 0, 0.0081), L = 1.602, DG = 0.3222. Solutions from Method P were (Nα)^{T} = (0.336, 0, 0, 0.173, 0, 0.164), β^{T} = (0.0220, 0, 0, 0.0182, 0, 0.0086), L = 1.487, DG = 0.3513.
Acknowledgments
Johan A. M. Van Arendonk is gratefully acknowledged for encouraging and giving P.B. the opportunity to visit J.A.W., and for giving useful comments on this manuscript. One author (J.A.W.) gratefully acknowledges the Ministry of Agriculture, Fisheries and Food (United Kingdom) for financial support. Jack C. M. Dekkers is acknowledged for giving very useful comments on this manuscript. This research was financially supported by the Netherlands Technology Foundation and was coordinated by the Life Sciences Foundation.
Footnotes

Communicating editor: T. Mackay
 Received March 16, 1998.
 Accepted November 17, 1998.
 Copyright © 1999 by the Genetics Society of America