Prediction of Genetic Contributions and Generation Intervals in Populations With Overlapping Generations Under Selection

Corresponding author: Piter Bijma, Animal Breeding and Genetics Group, Department of Animal Sciences, Wageningen Agricultural University, P.O. Box 338, Marijkeweg 40, 6700AH Wageningen, The Netherlands., piter.bijma{at}alg.vf.wau.nl (E-mail)

Communicating editor: T. MACKAY

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

A method to predict long-term 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 long-term genetic contribution as a linear regression on breeding value. Total genetic contributions of age classes are modeled using a modified gene flow approach and long-term 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 long-term genetic contributions of individual animals, as well as total contributions of age classes were obtained. Due to selection, offspring of young parents had an above-average breeding value. Long-term 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 long-term 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 well-known 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 (BICHARD et 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 long-term 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 (WRAY et 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 closed-form approximation for the prediction of long-term 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 long-term 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 long-term 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 long-term 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 long-term 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 long-term genetic contributions are also presented, but this is not the main item, as accurate predictions of genetic gain are already well established (e.g., VILLANUEVA et al. 1993 ).

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

Here we first describe the population structure that was used. Subsequently we describe the concept of long-term genetic contributions and the method of WOOLLIAMS et al. 1999 for the prediction of long-term 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 long-term 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 = ^cmax_k=1n_k and N_f = ^2cmax_k=cmax+1n_k, respectively. Using random mating, each sire was mated to d dams (), and each dam produced a total of n_o offspring (¹/₂n_o of each sex), so that the total number of offspring in a cohort equalled T = n_oN_f. Before reproductive age, the phenotype of individuals was recorded and a selection index was calculated. Because index weights were constant over time and no additional phenotypic information was included in the index at later ages, the index of individuals remained constant over time and the ranking of animals within categories remained unchanged over time. Within categories, individuals were ranked on the index, and the highest ranking n_k individuals were selected. The number of parents selected from each category was determined in advance and remained constant over time, as in conventional gene flow (HILL 1974 ). Selection on estimated breeding value across categories, which gives the highest genetic level of the offspring in the next generation (JAMES 1987 ), was not applied. The selection index was

where P is the phenotype of the individual, _FS is the mean of n_o full-sib records (including the individual), and _HS is the mean of n_od half-sib records (including the individual and its full sibs). This form was used by WRAY et al. 1994 and is convenient because the three sources of information are independent, which simplifies expressions such as the accuracy of selection. Note that mass selection is a special case of this index, where b₁ = b₂ = b₃. Different sets of index weights were chosen to allow for different selection strategies, i.e., for a varying emphasis on family information.

Basic approach for prediction of long-term genetic contributions:
This section introduces the concept of long-term genetic contributions. The long-term genetic contribution (r_i) of ancestor i in cohort t₁ is defined as the proportion of genes present in all individuals in cohort t₂ deriving by descent from i, where (t₂ - t₁) (WOOLLIAMS et al. 1993 ). In other words, the long-term 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 (long-term 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, long-term 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 (WRAY et al. 1994 )]. For an individual in category l, E(r_i(l)|A_i(l)) u_i(l) = _l + ß_l (A_i(l) - _l), where _l is the expected contribution of an average parent in category l, ß_l is the regression of the contribution of i on its breeding value (A_i(l)), and _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 is obtained directly from the number of parents: = (2N_x)^-1, (x = m, f; WRAY and THOMPSON 1990 ). For both discrete and overlapping generations, solutions for ß can be obtained from two regression models (WOOLLIAMS 1998 ; WOOLLIAMS et al. 1999 ): first, the regression of the number of selected offspring on the breeding value of the parent (), and second, the regression of the breeding value of selected offspring on the breeding value of the parent (). Both and can be computed on the basis of known parameters; a derivation is in APPENDIX A. Under equilibrium genetic parameters (BULMER 1971 ), regression coefficients (, ß, , ) are equal for the parental and offspring generation, allowing for the following closed form expression to compute ß instead of a recursive algorithm (WOOLLIAMS 1998 ):

Prediction of expected long-term genetic contributions in populations with overlapping generations:
This section describes the approach of WOOLLIAMS et al. 1999 to predict long-term genetic contributions for populations with overlapping generations. For ancestor i in category l, the expected long-term genetic contribution was predicted from u_i(l) = _l + ß_l(A_i(l) - _l). Predictions of genetic contributions are obtained using a modified gene flow matrix (G) of dimension 2c_max x 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 G₀, elements g⁰_kl represent the proportion of genes currently in category k that were in category l one time unit ago. In the modified gene flow matrix, elements g_kl of G represent the proportion of genes in the n_k selected individuals in category k that were contributed by parents in category l. (Contributed by a parent in category l refers to contribution via offspring that were born when the parent was in category l.) Because G represents the parental origin of the genes of selected individuals, it is affected by the degree of selection that is taking place, and this may vary with age. Because selected individuals may be born c_max years ago (and the age of parents at birth of offspring is relevant), G has a memory of c_max years, whereas G₀ has only 1 year memory.

Solutions for and ß were obtained from the basic equations (WOOLLIAMS et al. 1999 ),

(1)

(2)

where _kl is the regression coefficient of the selected number of offspring in category k on the breeding value of the parent in category l, and _kl is the regression of the breeding value of selected offspring in category k on the breeding value of the parent in category l. An intuitive understanding of Equation 1 and Equation 2 can be gained by noting that 2n_l^-1g_kln_k represents the average number of selected offspring in category k of an ancestor in category l. Therefore, in (1), _l is equal to ¹/₂ times the sum of the average contributions of all selected offspring. (The other ¹/₂ originates from the other parent.) In (1) it is implicitly assumed that the contribution of an average selected offspring (_k) is not dependent on the category of the parent (l). In (2), the first summation represents the change of contributions due to deviations of the selected number of offspring from the average. The second summation represents changes of genetic contributions of ancestors due to deviations of the breeding value of selected offspring of this ancestor from the average breeding value of selected contemporaries. In matrix form, combining Equation 1 and Equation 2 for all categories l (WOOLLIAMS et al. 1999 ),

(3)

(4)

where * denotes the element-by-element multiplication, T denotes the transpose of matrices, I is a 2c_max x 2c_max identity matrix, N is a 2c_max x 2c_max diagonal matrix of elements n_k, is a 2c_max x 2c_max matrix of elements _kl, is a 2c_max x 2c_max matrix of elements _kl, is a 2c_max vector of elements _l, and ß is a 2c_max vector of elements ß_l. Throughout the article, matrices follow the gene flow notation, i.e., rows represent offspring categories and columns represent parental categories. Prediction of genetic contributions using Equation 3 and Equation 4 is referred to as Method M in RESULTS.

Improved modified gene flow: A first-order correction to Equation 1 was derived by taking account of differences among average breeding values of parental subgroups present in the selected offspring (WOOLLIAMS et 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 (WOOLLIAMS et al. 1999 ),

(5)

where E[(A_(i)k - _k) given i has category l parent] is the expected breeding value of a selected offspring in category k descending from a category l parent, as deviation from the mean of selected contemporaries in category k. Substituting Equation 2 for ß, the resulting expression is (WOOLLIAMS et al. 1999 ),

(6)

where D is a 2c_max x 2c_max matrix of elements d_kl = E[(A_(i)k - _k) given i has category l parent] . Therefore, N is obtained as a right eigenvector of the 2c_max x 2c_max matrix [G^T + (G^T * D^T)(I - G^T * ^T)^-1 (G^T * ^T)] with an eigenvalue of one (WOOLLIAMS et al. 1999 ). Solutions for ß are still obtained from (4). Predictions of genetic contributions using Equation 4 and Equation 6 will be referred to as Method P in RESULTS.

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 long-term 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: _Lu_i = 1. The generation interval (in years) is therefore equal to the reciprocal of the total long-term genetic contribution per year, i.e., summed over all ancestors per year. In u_i(l) = _l + ß_l(A_i(l) - _l), the term ß(A_i(k) - _k) is zero on average, the sum of genetic contributions is therefore equal to ^2cmax_k=1n_kk, and generation interval was calculated as (WOOLLIAMS et al. 1999 ),

(7)

Generation intervals from this definition were compared to generation intervals defined as the average age of parents at birth of their offspring.

Deterministic prediction procedure:
Elements of Equation 3Equation 4Equation 5Equation 6Equation 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 (WOOLLIAMS et al. 1999 ).] Predictions of genetic contributions shown in RESULTS are based on BULMER's (1971) equilibrium parameters. A numerical example is in Appendix 1.

Phenotypic variance in year t was the sum of additive genetic variance and environmental variance, ²_p,t = ²_A,t + ²_E. Environmental variance was constant over time. Additive genetic variance in an unselected cohort born at year t was calculated as

where ²_A(m),t and ²_A(f),t are the between-sire and between-dam family additive genetic variance in unselected newborn offspring, and ²_A0 is the base generation additive genetic variance. Because genetic contributions are mainly determined in the first few generations, they are hardly affected by the rate of inbreeding. Therefore, no effect of inbreeding on the within-family variance was modeled.

Between-sire family additive genetic variance was calculated from

where 2g⁰_1l is the proportion of offspring descending from sires in category l (2g⁰_1l = ), _l is PEARSON 1903 variance reduction coefficient, _t is the accuracy of the index in year t (WRAY et al. 1994 ), µ_l,t is the average breeding value of selected sires in category l, and µ_(m),t is the average breeding value of all selected sires, i.e., µ_(m),t = ^cmax_l=12g⁰_1lµ_l,t. Between-dam family additive genetic variance was calculated in the same way. For the calculation of (µ_l,t-1 - µ_(m),t-1)², only differences between breeding values of selected individuals are important, and breeding values can be expressed relative to an arbitrary base. The genetic level of unselected animals at birth was taken as base here, and therefore, µ_l,t = (i_ltA,t - age(l)G_t), where i_l is the selection intensity in category l (not distinguishing between subgroups within categories and ignoring deviations from normality), and G_t is the rate of genetic gain in year t. (It is assumed here that the difference between consecutive age classes is equal to G from the last iteration, because this assumption decreases the number of iterations needed to reach equilibrium values that are not affected by the assumption.)

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)

where p_kl is the selected proportion in the subclass descending from parents in category l, I_k is the index truncation point common for all offspring in category k, _I,l is the standard deviation of the selection index of individuals descending from parents in category l, denotes the cumulative normal density, and _(x) is twice the regression of the index of the offspring on the breeding value of the parent of sex x (x = m, f; WRAY et al. 1994 ); i.e., the term ¹/_2(x)µ_(x)l represents the average index value of offspring descending from parents in category l. A solution for the common truncation point was obtained using the algorithm RIDDR ROOT from Numerical Recipes (PRESS et al. 1992 ). Elements of G were derived from

Elements of D are d_kl = E[(A_(i)k - _k) given i has category l parent] and were calculated as (omitting subscript t for simplicity)

with _k = _l2g_kl[µ_l + i_klAl] calculated separately for each sex; where i_kl is the selection intensity in subclass kl, and _Al is the additive genetic variance among unselected offspring descending from parents in category l.

Elements of were calculated as _kl = (1 - _{k(x) A} x ^-1_I). Elements of were calculated as _kl =i_k(x)^-1_I (see Appendix 1). A general procedure to derive and is in WOOLLIAMS et al. 1999 .

As described in the section on prediction of long-term genetic contributions, 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 , then nx will also be an eigenvector of A with the same eigenvalue . With the same eigenvalue, therefore, different eigenvectors can be obtained from Equation 3 or Equation 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 long-term genetic contribution of the categories (n_kk): L = . And second, was scaled so that ^2cmax_k=1n_kk = L^-1, i.e., is defined per year, and by definition the generation interval is the time in which contributions sum to unity.

Using E(G) = ^2cmax_k=1n_kE[r_i(k)a_i(k)], where a_i(k) is the Mendelian sampling value of i (WOOLLIAMS and THOMPSON 1994 ), genetic gain was predicted from E(G_t) = ²_A0[_w^-1_I x ^2cmax_k=1n_kki_k + ^2cmax_k=1n_kß_k(1 -_kwA^-1_I)]_t, where _w is the regression of the index on the Mendelian sampling effect of the individual. A derivation is in Appendix 1.

Stochastic simulation:
To draw inferences on the accuracy of predicted genetic contributions, the breeding scheme described in the Population model section was simulated stochastically and genetic contributions were estimated from simulated data. A noninbred and unselected base population of the appropriate family structure was generated. Breeding values of base population animals were taken from N(0, ²_A0), and environmental values were from N(0, ²_E). Within categories, individuals were ranked on the index, and the highest ranking n_k individuals were selected from the kth category. Breeding values of offspring were obtained as ¹/₂A_m + ¹/₂A_f + a, where A_m, A_f, and a are the sire and dam breeding values and the Mendelian sampling value. No effect of inbreeding on the Mendelian sampling variance was simulated, i.e., a ~ N(0, ²_A0).

For the calculation of genetic contributions, an ancestor cohort t₁ 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 t₂ was chosen. Convergence time of genetic contributions (t₂ - t₁) was approximately equal to 7c_max. The long-term genetic contribution of ancestor i in category l in cohort t₁ to individuals in cohort t₂ was obtained by summing contributions via all pedigree paths leading from i to individuals in t₂, r_i(l) = T^{-1 T}_j=1r_i(l),j, where r_i(l),j is the contribution to individual j in cohort t₂. r_i(l),j was calculated as _paths^-1, where is the total number of animals (including i and j) in a pedigree path from i to j.

Genetic contributions were analyzed using the model r_i(l) = _l + ß_l(A_i(l) - _l) + e_i(l) . was estimated as _l = n^-1_l ^nl_i=1r_i(l) and ß was estimated as _l = ^nl_i=1r_i(l) . Asymptotic rate of genetic gain was calculated as G = , where G_t is the average breeding value of all animals born in cohort t. Generation interval was calculated as L = . Results were averaged over 500 replicates and standard errors were calculated from the variance among replicates.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

In this section, a comparison is made between results from conventional gene flow (Method C; HILL 1974 ), simple modified gene flow (Method M, Equation 3 and Equation 4), and improved modified gene flow (Method P, Equation 4 and Equation 6), for mass and sib-index selection.

Mass selection
Accuracy of : Table 1 shows long-term genetic contributions of categories (n_kk) 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.

View this table: In this window In a new window   Table 1. Genetic contributions of categories (nkk) under mass selection

Results from Method C are independent of heritability (h²₀), but results from Method M, Method P, and from simulation are not. For h²₀ = 0.01, results from all methods are practically identical because heritable effects play a minor role in that case. For higher heritabilities, Method C shows considerable overestimates of contributions from 3-yr-old dams (n₆₆), whereas Methods M and P are significantly closer, and from these, Method P is most accurate. For high heritabilities (>0.6), absolute differences between Method P and simulation are roughly only 10% of the errors from Method C, and for this particular scheme in the opposite direction. The large differences between Method C and simulation are partly caused by the distribution of parents across age classes in Table 1. Because most dams are selected from the oldest category, offspring from these dams will have a low breeding value, which will reduce their genetic contribution. When parents are selected across age classes, differences between Method C and simulation will be much smaller (see DISCUSSION).

Comparing Methods M and P to simulation results shows that the first-order correction improves the accuracy of the predicted long-term 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 h²₀ = 0.99, probably due to deviations from normality for this extreme case, which is of little practical importance).

Accuracy of ß: Table 2 shows the regression coefficients of contributions on breeding values (ß), 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 3-yr-old females, i.e., ß₄ and ß₆. In Method C, the effect of selective advantage is not modeled, i.e., ß 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 G₀. 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 1) 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 G₀, 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.

View larger version (23K): In this window In a new window Download PPT slide   Figure 1. Predicted long-term genetic contributions from Method P of 1-yr-old females as a proportion of the total contribution of females n44/(n44 + n66), as a function of heritability, for S1: N = diag{20,0,0,30,0,10) and S2: N = diag{20,0,0,10,0,30}, for no = 4 or no = 20 tested offspring per dam, and mass selection (b1 = b2 = b3).

View larger version (21K): In this window In a new window Download PPT slide   Figure 2. Predicted generation interval (L) from Method P, as a function of heritability, for S1: N = diag{20,0,0,30,0,10) and S2: N = diag{20,0,0,10,0,30}, for no = 4 or no = 20 tested offspring per dam, and mass selection (b1 = b2 = b3).

View this table: In this window In a new window   Table 3. Rate of genetic gain (G) and generation interval (L) under mass selection

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 : The effect of heritability and selection intensity on average genetic contributions of categories (n_kk) was studied using Method P. Figure 1 shows the predicted long-term genetic contribution of 1-yr-old females as a proportion of the total contribution of females (n₄₄/(n₄₄ + n₆₆)), for two different breeding schemes and for two selection intensities. The breeding schemes were S₁: N = diag{20,0,0,30,0,10} and S₂: 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 S₁ and S_2, males are selected from a single age, and L is directly related to n₄₄/(n₄₄ + n₆₆). Results from Method C are identical to results for h² = 0.

Figure 1 clearly shows an increased contribution of 1-yr-old females when heritability increases, which is due to an increased selective advantage of offspring descending from 1-yr-old dams when h²₀ increases. As heritability increased from 0.2 to 0.8, genetic gain per year increased from 0.232 to 0.977 units _p for N = diag{20,0,0,10,0,30} and n₀ = 20. Consequently, the difference between average breeding values of offspring from 1- and 3-yr-old dams increased from 0.277 to 1.153. This selective advantage resulted in an increased proportion of offspring selected from 1-yr-old dams when h²₀ increased. When h²₀ increased from 0.2 to 0.8, it showed that among the 1-yr-old selected females, the proportion descending from 1-yr-old dams increased from 0.386 to 0.894. [These proportions were determined from the G matrix (not shown).]

The relative long-term genetic contribution of 1-yr-old 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 1-yr-old dams, in the same way as when h²₀ increases, but also due to a decreased overall selected proportion moving the common truncation point of subclasses to the right. When a common truncation point for two normal distributions with different means is moved to the right, the smaller upper-tail probability of the two will decrease more rapidly than the larger upper-tail probability, due to the nonlinear relation between truncation point and selected proportion, therefore decreasing the relative contribution of 3-yr-olds. This effect can be illustrated by comparing the relative contribution of 1-yr-old females between schemes with different selection intensities at the same G, because with the same G the difference between mean breeding values of 1- and 3-yr-old dams will be the same. For N = diag{20,0,0,10,0,30}, n_o = 20 and h²₀ = 0.4, G was 0.4854, and the same G can be obtained with identical N, but with n_o = 4 and h²₀ = 0.77. However, the relative contribution of 1-yr-old females differed considerably: 0.685 for n_o = 20 compared to 0.540 for n_o = 4 (see Figure 1), mainly due to different selection intensities.

Effect of selection intensity on ß: Figure 3 shows the relation between selection intensity and ß 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).

View larger version (16K): In this window In a new window Download PPT slide   Figure 3. Predicted regression coefficients of long-term genetic contributions on breeding values (ß) from Method P as a function of selection intensity (i), for age class 1 males (ß1), age class 1 females (ß4), and age class 3 females (ß6), for h20 = 0.4, N = diag{20,0,0,20,0,20}, and mass selection (b1 = b2 = b3).

Figure 3 shows an increase in ß₁ and ß₄ with increasing selection intensity. On average, ß is expected to increase with selection intensity because the regression of selected number of offspring on breeding value () increases with selection intensity (see Appendix 1) and ß is positively related to (see Equation 2), explaining the trend for ß₁ and ß₄. For ß₆ the increase with selection intensity is counteracted by the reduced total contribution of 3-yr-old dams (see Figure 1). For other heritabilities (results not shown) the relation between ß and selection intensity was similar.

Effect of heritability on ß: Figure 4 shows the relation between ß and heritability using Method P. For h²₀ = 0, ß₄ = ß₆ = ß₁, which is to be expected from (2) when selection intensity is equal for all categories and g_kl = g⁰_kl because h²₀ = 0. When h²₀ increases, genetic gain increases, resulting in a higher proportion of selected offspring descending from 1-yr-old parents, i.e., for all k, g_k4 > g_k6 for h²₀ > 0. When g_k4 > g_k6 and selection intensity is equal for all categories, it can be inferred from (2) that ß₄ > ß₆ as in Figure 4.

View larger version (18K): In this window In a new window Download PPT slide   Figure 4. Predicted regression coefficients of long-term genetic contributions on breeding values (ß) from Method P as a function of heritability (h20), for age class 1 males (ß1), age class 1 females (ß4), and age class 3 females (ß6), for N = diag{20,0,0,20,0,20}, no = 8 tested offspring per dam, and mass selection (b1 = b2 = b3).

It is a general conclusion for mass selection, therefore, that ß of younger categories will increase with h²₀, whereas ß of older categories will decrease with h²₀. The interpretation of this relation is, that under mass selection the contributions of young animals will increasingly be determined by their breeding value when h²₀ increases, whereas for older animals the effect of breeding value on contributions will decrease with increasing h²₀. An intuitive way of looking at this is, that for influential animals (which are young animals when h²₀ is high) a change of breeding value gives a larger (absolute) change of genetic contributions than it does for unimportant animals. The same reasoning holds for the relation between ß and selection intensity, explaining the different trend of ß₄ and ß₆ in Figure 3.

The regression coefficient for 1-yr-old males (ß₁) shows only minor variation with h²₀ because males are selected from a single category in Figure 3. Therefore, category 1 always contributes 50% of the genes of selected offspring (g₁₁ = g₄₁ = g₆₁ = 0.5) regardless of heritability, and variation of ß₁ with h²₀ is only due to variation in and .

Selection on a sib index
Long-term genetic contributions of categories (n_kk) 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 and , resulting in different predictions for ß. Results for a sib index, therefore, focus on ß, though will also differ from results for mass selection.

Accuracy of ß: 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 long-term responses are generally different from classical index weights.

For positive weight on family information, Table 4 shows ß from Method P and from simulation for N = diag{20,0,0,10,0,30}, b₁ = 1, b₂ = 1.5, and b₃ = 2 (i.e., I = P + _FS + _HS). In Table 4, Method P shows the same trend as simulation results, but tends to slightly overestimate regression coefficients for 1-yr-old parents (ß₁ and ß₄). Predictions of (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 ß from Method P and from simulation, for N = diag{20,0,0,10,0,30}, b₁ = 1, b₂ = 0.5, and b₃ = 0 (i.e., I = P - _FS - _HS). In Table 5, Method P shows the same trend as simulation results and is accurate. Predictions for (results not shown) were very accurate, i.e., within ±3 SE with 500 replicates in the simulation.

Effect of index weights on ß: Figure 5 shows the effect of a varying emphasis on family information in the selection index on the regression coefficients of long-term genetic contributions on breeding values, for 1-yr-old male parents (ß₁), from Method P (lines), and from simulation (markers) for N = diag{20,0,0,20,0,20}. For this scheme, ß₁ gives a good impression of the average level of ß, 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 b₁ = 1, b₂ = b₃ = 0, representing complete within-family selection, to b₁ = 1, b₂ = 2, b₃ = 2, which is identical to I = P + _FS.

View larger version (20K): In this window In a new window Download PPT slide   Figure 5. Predicted and simulated regression coefficients of long-term genetic contributions on breeding values for 1-yr-old males (ß1) from Method P, for different index weighting factors (b) and a range of heritabilities (h20). For N = diag{20,0,0,20,0,20} and no = 8. Lines indicate predictions, markers indicate results from simulation.

For within-family selection, ß 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 and ß) are identical to results from Method C.

When index weights on family information increased, ß₁ increased because selection of offspring is increasingly affected by the parental breeding value. Similar relations between the average level of ß 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 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

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

This article has studied in detail two methods proposed by WOOLLIAMS et al. 1999 for the prediction of long-term genetic contributions of individuals in selected populations with overlapping generations. The methods enable accurate predictions of long-term 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 long-term 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 long-term genetic contributions [by including ß_k(A_i(k) - _k) in the model for expected contributions], whereas conventional gene flow only enables prediction of average genetic contributions (i.e., assuming ß = 0). For the prediction of rates of inbreeding it is crucial to account for the effect of selection between individuals (WRAY et 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 = , i.e., the generation interval is the time in which long-term 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 L₀ = , where ₀ are contributions obtained from conventional gene flow. Generation interval based on contributions to selected offspring only (L₁), i.e., the average age of parents of selected offspring, can be obtained from the modified gene flow matrix G (see Appendix 1) and was close to results from simple modified gene flow. When genetic gain is made and selective advantage is inherited, generation interval based on long-term genetic contributions is shorter than both L₀ and L₁, because selective advantage is partly passed on to more distant offspring.

Whereas L₀ and L₁ 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 long-term 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 long-term 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 L₀ and L₁.

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 Table 1 and Table 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 h²₀ =0.5, n_o = 4, N_m = 20, N_f = 40, truncation selection across categories resulted in N = diag{18,2,33,7}, predicted generation interval from conventional gene flow was 1.138 and from modified gene flow was 1.129 (simulation: L = 1.130). An advantage of modified gene flow is that it gives accurate predictions of generation interval for any arbitrary distribution of parents across categories, and it is not limited to truncation selection across categories.

In the present article, the within-family 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. Long-term 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 long-term genetic contributions (WOOLLIAMS et 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 long-term 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 long-term 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 well-known 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.

	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.

Manuscript received March 16, 1998; Accepted for publication November 17, 1998.

	APPENDIX 1

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 APPENDIX 1 LITERATURE CITED

Derivation of :
Elements _kl are obtained as _kl = p^-1_{k x}b_SkIk, where ¹/_2x is the regression of the index of the offspring on the breeding value of the parent of sex x (WRAY et al. 1994 ), and b_SkIk is the regression of the selection score (selected or not selected, i.e., S = 1 or 0) of the offspring on its index. In Equation 2, _kl is expressed per selected offspring, whereas S is an expression per selection candidate, the difference being on average a factor p^-1_k. Tau is obtained as _x = 2 resulting in _m = b₃, _f = b₂ (1 - ) + . From a result of Robertson (appendix in DEMPSTER and LERNER 1950 ), b_SkIk = p_ki_k^-1_I. Resulting expressions are _kl = ¹/₂ b₃i_k^-1_I for male parents and _kl = (b₂(1 - ) + )i_k^-1_I for female parents.

Derivation of :
Elements _kl are obtained as _kl = *, where * denotes (co)variances after selection of the offspring. Using COCHRAN 1951 , ^*_AB = _AB - () for the calculation of Cov(A,B) after selection on I gives

where Cov(A_i(l),A_j(k)) = ^2*_A(x)l, Cov(A_i(l),I_j(k)) = _x^2*_A(x)l, Cov(A_j(k),I_j(k)) = _AI, Var(I_j(k)) = ²_I, ^2*_A(x)l is the additive genetic variance among selected parents, and x denotes the sex of parent i. Assuming that Var(A_i(l)) is little affected by selection of the offspring, i.e., Var(A_i(l))* = ^2*_A(x)l, the resulting expression becomes _kl = (1 - _kxA^-1_I).

	APPENDIX 1