Predicting the range of inbreeding depression of inbred lines in cross-pollinated populations

Abstracts

The objectives of this paper were to derive the genetic variance of inbreeding depression (<img SRC="Image482.gif" WIDTH="48" HEIGHT="33"> ) and to predict the range of inbreeding depression (RID) in cross-pollinated populations. The variance of inbreeding depression is a function of the genetic variances related to dominance effects (<img SRC="Image483.gif" WIDTH="31" HEIGHT="33">, D2, and <img SRC="Image484.gif" WIDTH="21" HEIGHT="25">), and of the inbreeding coefficients of the two generations in which inbreeding depression is measured (Ft and Fg). The results showed that the higher the level of dominance of a trait, the higher the variance of inbreeding depression. The magnitudes of <img SRC="Image485.gif" WIDTH="48" HEIGHT="33">were expected to be lower in improved (mean gene frequencies = <img SRC="Image486.gif" WIDTH="16" HEIGHT="25">> 0.6) and in unimproved (<img SRC="Image487.gif" WIDTH="16" HEIGHT="25"> < 0.4) populations, than in composite populations (<img SRC="Image487.gif" WIDTH="16" HEIGHT="25"> <FONT FACE="Symbol">»</font> 0.5). Data from a maize population used to illustrate the study showed that the range of inbreeding depression in the S<FONT FACE="Symbol">¥</font> generation of selfing was from 48.7% to 85.3% for grain yield, and from 13.9% to 24.5% for plant height. A mating design outlined to estimate the genetic variance of inbreeding depression, the range of inbreeding depression, and of the range of inbred lines is presented.

Os objetivos deste artigo foram derivar a variância genética da depressão por endogamia (<img SRC="Image505.gif" WIDTH="48" HEIGHT="33">) e predizer a amplitude da depressão por endogamia (RID) dentro de populações de polinização cruzada. A variância da depressão por endogamia é função das variâncias genéticas relacionadas aos efeitos de dominância (<img SRC="Image493.gif" WIDTH="31" HEIGHT="33">, D2 e <img SRC="Image520.gif" WIDTH="21" HEIGHT="25">) e dos coeficientes de endogamia das duas gerações em que é medida a depressão por endogamia (Ft e Fg). Os resultados mostraram que quanto maior o nível de dominância de um caráter maior é a variância da depressão por endogamia, e que as magnitudes de <img SRC="Image505.gif" WIDTH="48" HEIGHT="33">devem ser menores em populações melhoradas (<img SRC="Image500.gif" WIDTH="16" HEIGHT="25"> > 0,6) e não melhoradas (<img SRC="Image500.gif" WIDTH="16" HEIGHT="25"> < 0,4) que em compostos (<img SRC="Image487.gif" WIDTH="16" HEIGHT="25"> <FONT FACE="Symbol">»</font> 0,5). Dados de uma população de milho utilizada como exemplo mostraram que a amplitude da depressão por endogamia na geração S<FONT FACE="Symbol">¥</font> de autofecundação é de 48,7% a 85,3% para produção de grãos e de 13,9% a 24,5% para altura da planta. Um delineamento apropriado para estimar a variância da depressão por endogamia, a sua amplitude e a amplitude de linhagens endogâmicas é apresentado.

Predicting the range of inbreeding depression of inbred lines in cross-pollinated populations

C.L. Souza Jr. ¹ and J.S.C. Fernandes ²

¹Departamento de Genética, ESALQ/USP, Caixa Postal 83, 13400-970 Piracicaba, SP, Brasil.

Send correspondence to C.L.S.Jr.

²Departamento de Genética, UFPR, Caixa Postal 19071, 81531-970 Curitiba, PR, Brasil.

ABSTRACT

The objectives of this paper were to derive the genetic variance of inbreeding depression () and to predict the range of inbreeding depression (RID) in cross-pollinated populations. The variance of inbreeding depression is a function of the genetic variances related to dominance effects (, D₂, and ), and of the inbreeding coefficients of the two generations in which inbreeding depression is measured (Ft and Fg). The results showed that the higher the level of dominance of a trait, the higher the variance of inbreeding depression. The magnitudes of were expected to be lower in improved (mean gene frequencies = > 0.6) and in unimproved ( < 0.4) populations, than in composite populations (» 0.5). Data from a maize population used to illustrate the study showed that the range of inbreeding depression in the S¥ generation of selfing was from 48.7% to 85.3% for grain yield, and from 13.9% to 24.5% for plant height. A mating design outlined to estimate the genetic variance of inbreeding depression, the range of inbreeding depression, and of the range of inbred lines is presented.

INTRODUCTION

Inbreeding depression depends on both the level of dominance of a trait and on the inbreeding coefficient; the higher the level of dominance of a trait the higher the inbreeding depression. The rate of inbreeding depression has a negative linear relationship with the coefficient of inbreeding for most traits (Falconer, 1989; San Vicente and Hallauer, 1993; Benson and Hallauer, 1994).

The major objective in breeding programs of several cross-pollinated species has been to develop hybrids from inbred lines, e.g., maize, sunflower, and sugar beet. Thus, highly inbred lines are obtained, usually by selfing, and subsequently crossed to produce hybrids (Miller, 1987; Smith, 1987; Hallauer, 1990). To be commercially useful, inbred lines must have outstanding performances themselves and in combining ability. Therefore, selecting lines based on their own performance during the generations of inbreeding is a standard procedure ( Hallauer, 1990).

The rates of inbreeding depression vary among populations of the same species, and populations less sensitive to inbreeding depression are selected as a potential source of outstanding inbred lines (Vianna et al., 1982; Lima et al., 1984). Inbreeding depression also is expected to vary within populations, i.e., genotypes of a population have different mean phenotypic values under inbreeding. Thus, inbred lines selected on their own performance are those whose parents were less sensitive to inbreeding depression.

Despite its importance, no information is available on the genetic variability of inbreeding depression within populations. Thus, the purpose of this paper was threefold: 1) to derive the genetic variance of inbreeding depression and present a mating design to estimate it; 2) to present procedures for predicting the range of inbreeding depression and of inbred lines in any generation of selfing; 3) to present an example for maize with reported experimental results.

METHODS

Consider a random mating population in both Hardy-Weinberg and linkage equilibrium. Excluding epistasis, the genetic variance of this population in the tth generation of selfing is (Cockerham, 1983): = (1 + Ft) + (1 - Ft) + 4FtD1 + FtD2 + Ft(1 - Ft), where and are additive and dominance variances, D1 is the covariance of additive and homozygous dominance effects, D2 is the variance of homozygous dominance effects, is the sum of squared inbreeding depression effects, and Ft is the inbreeding coefficient in the tth generation of selfing.

Let Xt and Xg be the means of a trait in the tth and gth (g > t) generations of inbreeding, with Ft and Fg as their inbreeding coefficients, respectively. The inbreeding depression being measured as ID = (Xt - Xg), its variance is = + - 2CovXt,Xg where and are the variances in the tth and in the gth generations of inbreeding, respectively, and CovXt,Xg is the covariance of individuals in the tth and in the gth generations. Following Cockerham’s (1983) notation, the components of the genetic variance of inbreeding depression could be expressed as = Cttt, = Ctgg, and CovXt,Xg = Cttg, where Cttt is the genetic variance in the tth generation; Ctgg is the genetic covariance among relatives in the gth generation with the last common ancestor in the tth generation, or the genetic variance among progenies in the gth generation stemming from a common ancestor in the tth generation; and Cttg is the genetic covariance between a relative in the tth generation and the other one in the gth generation.

With Cttt = , Ctgg = (1 + Ft) + [(1 - Fg)2/(1 - Ft)] + 2(Ft + Fg)D1 + [Ft + (Fg - Ft)2/2(1 - Ft)]D2 + [Ft(1 - Fg)2/(1 - Ft)] , and Cttg = (1 + Ft) + (1 - Fg) + (3Ft + Fg)D1 + FtD2 + Ft(1 - Fg) , (Cockerham, 1983), the genetic variance of inbreeding depression in the gth generation was derived as = Cttt + Ctgg - 2Cttg.

Expected values of the variances of inbreeding depression and of the ratio /D2 for several types of populations were obtained by a simulation procedure, assuming the population gene frequencies fit a beta distribution (Souza Jr., 1985, 1987, 1993). Let p and q refer to the frequencies of favorable and unfavorable alleles, and a and d refer to half the difference between homozygotes and the dominance effect, respectively; thus = (2pqd)2 and D2 = 4pq(q - p)2d2 (Cockerham and Matzinger, 1985). The expected values of variances were obtained for populations with favorable gene frequencies ranging from = 0.1 to = 0.9 as E() = f pdp, where is or D2 and fp is a Beta density function.

The mean value of a trait in a random mating population is S0 = m + a + d, where m is the general mean, a = (pi - qi)ai and d = 2piqidi are the contributions of the homozygous and heterozygous loci, respectively, to the mean value (Gardner, 1965). In the gth generation of selfing the mean value can be expressed as Sg = m + a + (1/2)g d; thus, the mean of all pure lines extracted at random from the population is S¥ = m + a, and the mean of random lines from one generation of selfing is S1 = m + a + (1/2)d. From the means of the S0 and S1 generations one can estimate S¥ = 2S1 - S0, and d = 2(S0 - S1), and the mean values of any generation of selfing can be estimated as Sg = S¥ + (1/2)g d. Inbreeding depression in the gth generation of inbreeding was estimated as IDg = Sg - S0, and in percentage as IDg% = (IDg/S0)100; the range of inbreeding depression in the gth generation was estimated as RIDg = IDg ± 2sIDg.

Data from Iowa Stiff Stalk Synthetic (BSSS) maize population reported by Hallauer and Miranda (1988) were used to illustrate this study. The mean grain yield values of non-inbred S0 = 168.60 g/plant, and of the first generation of selfing S1 = 112.10 g/plant, and the average estimate of dominance variance = 179.0 were used. For plant height the following values were used: S0 = 194.10 cm/plant, S1 = 175.50 cm/plant, and = 20.0. Estimates of D2 variances were not available, so they were obtained from the ratio /D2 = 1.50 for a population whose average frequencies of favorable alleles is = 0.6 from the simulation. Then, it was assumed that the frequencies of favorable alleles in the BSSS population are greater than 0.5, as reported by Hallauer et al.(1983). From this ratio, estimates of D2 used were 119.0 and 13.0 for grain yield and plant height, respectively.

RESULTS AND DISCUSSION

The genetic variance of inbreeding depression is:

= [(Fg - Ft)2/(1 - Ft)][ + (1/2)D2 + Ft ];

therefore is a function of the genetic variances related to dominance effects (, D2 and ) and of the inbreeding coefficients in which inbreeding depression is measured (Ft and Fg, Fg > Ft). For populations developed from the cross of two highly inbred lines, e.g., F2 populations, we have D2 = 0 and = (Cockerham, 1983) and = [(Fg - Ft)2 (1 + Ft)/(1 - Ft)] .

Usually, inbreeding depression is measured in relation to the non-inbred population (Ft = 0), which reduces the variance of inbreeding depression to:

= [ + (1/2)D2],

and for F2 populations we have:

= ,

and is a function of the square of inbreeding coefficient of the gth generation of inbreeding (Fg) and of the variance of dominance effects ( and D2). The genetic variance of inbreeding depression of F2 populations does not depend on the variance of homozygous dominance effects. However, this does not mean that of the F2 is lower than for other types of populations, because the magnitudes of dominance variances arising from F2 populations are expected to be higher than other types of populations, since the frequencies of all segregating alleles in F2 populations are 0.5 (Falconer, 1989).

The ratios of /D2 for populations defined by average frequencies of favorable alleles ranging from 0.1 to 0.9, depicted in Figure 1, showed that D2 is greater than for populations with < 0.3 and > 0.7, and that > D2 between these values. Then, one could expect that D2 > in unimproved and improved populations, and that the contribution of the D2 variance to will be more expressive for these types of populations.

Figure 1 - Ratios of /D2 for average frequencies of favorable alleles ranging from 0.1 to 0.9.

For highly inbred lines (Fg = 1), and taking the non-inbred population as a reference (Ft = 0), the variance of inbreeding depression reduces to:

= + (1/2)D2,

and the expected variances for populations defined by average frequencies of favorable alleles ranging from 0.1 to 0.9 and for two levels of dominance (d = 2 and d = 4), depicted in Figure 2, showed that the lower the level of dominance of a trait the lower the variance of inbreeding depression, and irrespective of the level of dominance they follow the same pattern of variation. The variance of inbreeding depression increases from = 0.1 to = 0.4 and decreases from = 0.6, reaching a plateau between = 0.4 and = 0.6, this variation being more expressive for traits with higher levels of dominance. Thus, one would expect that of improved and unimproved populations will be lower than in populations whose average frequencies of favorable alleles lie between these intervals, which may be representative of composite populations (» 0.5). These results indicate that the higher the level of dominance the higher the range of inbreeding depression, and inbred lines extracted from unimproved and improved populations will have similar and lower ranges of inbreeding depression than composite populations.

Figure 2 - Genetic variances of inbreeding depression () of highly inbred lines (Fg = 1.0) for average frequencies of favorable alleles ranging from 0.1 to 0.9, and for two levels of dominance (d = 2 and d = 4).

For the one locus-two alleles model, the expected change in inbreeding depression in populations under selection is DID = -2FSiDpi(qi - pi)di, where Dpi is the change in the allele frequency at the ith locus due to selection, and for complete dominance inbreeding depression will decrease for > 0.5 (Souza Jr., 1985). Then, improved ( > 0.5) populations are expected to be less sensitive to inbreeding depression than unimproved populations ( < 0.5) and composite populations (» 0.5). Although the range of inbreeding depression will be lower in improved than in composite populations, the lower inbreeding depression in improved populations assures that the mean values of the lines in the upper limit of the range will be higher than in composite populations. San Vicente and Hallauer (1993) reported that the rates of inbreeding depression of hybrids produced from old and new lines are quite similar, whereas Benson and Hallauer (1994) reported that the rates of inbreeding depression of the BSSS maize population under recurrent selection have decreased in the advanced cycles of selection for most of the traits.

The mean values and the genetic variances of inbreeding depression for several generations of selfing for grain yield (GY) and plant height (PH) for the BSSS maize population showed that the expected mean values of random highly inbred lines are 55.60 g/plant and 156.90 cm high; and the genetic variances of inbreeding depression are (238.50) and (26.50) for GY and PH, respectively

Table I

(

Table II

(

Table III

(

Table I
- Mean values and genetic variances of inbreeding depression (

Generations Sg	Inbreeding coefficient Fg+	Mean values
		GY	PH	GY	PH
S0 S1 S2 S3 S4 S5 S6 S¥	0.00 0.50 0.75 0.88 0.94 0.97 0.98 1.00	168.6 112.1 83.9 69.7 62.7 59.1 57.4 55.6	194.1 175.5 166.2 161.6 159.2 158.1 157.5 156.9	0.0 59.6 134.2 182.6 209.6 223.8 231.1 238.5	0.0 6.6 14.9 20.3 23.3 24.9 25.7 26.5

+Rounded values.

Table II
- Mean values (ID) and ranges (RID) of inbreeding depression for grain yield (GY) in g/plant and in percentage for several generations of selfing (Sg) for the BSSS maize population.

Generations Sg	ID (g/plant)	RID (g/plant)		ID%	RID (%)
		Lower	Upper		Lower	Upper
S1 S2 S3 S4 S5 S6 S¥	56.5 84.8 98.9 105.9 109.5 111.2 113.0	41.1 61.6 71.9 77.0 79.5 80.8 82.1	71.9 107.9 125.9 134.9 139.4 141.6 143.9	33.5 50.3 58.6 62.8 64.9 66.0 67.0	24.4 36.5 42.6 45.7 47.2 47.9 48.7	42.7 64.0 74.7 80.0 82.7 84.0 85.3

ID = S0 - Sg; ID% = (ID/S0)100; RID = ID ± 2s_GID; RID% = (RID/ S0)100.

Table III
- Mean values (ID) and ranges (RID) of inbreeding depression for plant height (PH) in cm/plant and in percentage for several generations of selfing (Sg) for the BSSS maize population.

Generations Sg	ID (cm/plant)	RID (cm/plant)		ID%	RID (%)
		Lower	Upper		Lower	Upper
S1 S2 S3 S4 S5 S6 S¥	18.6 27.9 32.6 34.9 36.0 36.6 37.2	13.5 20.2 23.6 25.2 26.1 26.5 27.0	23.7 35.6 41.6 44.5 46.0 46.8 47.5	9.6 14.4 16.8 18.0 18.6 18.9 19.2	7.0 10.4 12.1 13.0 13.4 13.6 13.9	12.2 18.4 21.4 23.0 23.7 24.1 24.5

ID = S0 - Sg; ID% = (ID/S0)100; RID = ID ± 2s_GID; RID% = (RID/S0)100.

Notice that for Ft = 0, the variance of inbreeding depression is directly related to the square of the inbreeding coefficients of the lines (). The increases with Fg, but the increments are not linear, increasing sharply from S1 to S3 generation of selfing and decreasing thereafter. For example, the ratios (/)100 are 25%, 56%, and 77% for S1, S2, and S3 generations,respectively, whereas for S4, S5, and S6 generations the ratios are 88%, 94%, and 97%, respectively, for the two traits studied. The ranges of inbreeding depression followed the same pattern, as expected.

Estimates of variance of inbreeding depression can be obtained by producing unrelated single-crosses from highly inbred lines extracted at random from an open-pollinated population, subsequently producing S1 progenies by self-pollinating the single-crosses, then evaluating the single-crosses and their S1 progenies in replicated experiments following a split-block design. Analyses of variance of the single-crosses, of the S1 progenies, and of covariance analyses of single-crosses and S1 progenies are required to provide estimates of genetic variances of SC () and S1 progenies (), and of genetic covariances of SC and S1 (CovSC,S1). Thus we have,

= + ;

= + (1/4) + D1 + (1/8)D2, and

CovSC,S1 = + (1/2) + (1/2)D1.

The genetic variance of inbreeding depression of the S1 progenies is: = + - 2CovSC,S1; therefore, = (1/4) [ + (1/2)D2]. As we have shown, for Ft = 0 the genetic variance of inbreeding depression in the gth generation of inbreeding is = [ + (1/2)D2]; then, the genetic variance of inbreeding depression in the gth generation of inbreeding can be estimated as = 4.

For plant breeders, it would also be important to obtain estimates of the genetic variance and the range of the traits of the highly inbred lines. This could be accomplished by evaluating the highly inbred lines used to develop the single-crosses for the mating design previously described. However, it is difficult to obtain reliable estimates from the evaluation of highly inbred lines because of the genetic load that often is high in cross-pollinated populations. Also, highly inbred lines are more sensitive to environmental variation than partly inbred lines or non-inbred plants, which may interfere in the estimates of genetic variances (Falconer, 1989). The mating design presented also could provide these estimates as follows; the genetic variance of highly inbred lines is (Cockerham, 1983):

= + 4D1 + D2,

and can be estimated as:

= 2(4 + - 4CovSC,S1),

and the range of the mean values of the inbred lines can be estimated as RIL = S¥ ± 2.

The genetic model did not take into account linkage and/or Hardy-Weinberg disequilibrium and epistasis, and it was restricted to a one locus-two alleles model. Furthermore, the study was based on the assumption that the population gene frequencies fit a Beta distribution. Thus, it would be useful to estimate genetic variances of inbreeding depression and of highly inbred lines from experimental studies following the procedures outlined in this paper.

Publication supported by FAPESP.

RESUMO

Os objetivos deste artigo foram derivar a variância genética da depressão por endogamia () e predizer a amplitude da depressão por endogamia (RID) dentro de populações de polinização cruzada. A variância da depressão por endogamia é função das variâncias genéticas relacionadas aos efeitos de dominância (, D2 e ) e dos coeficientes de endogamia das duas gerações em que é medida a depressão por endogamia (Ft e Fg). Os resultados mostraram que quanto maior o nível de dominância de um caráter maior é a variância da depressão por endogamia, e que as magnitudes de devem ser menores em populações melhoradas ( > 0,6) e não melhoradas ( < 0,4) que em compostos (» 0,5). Dados de uma população de milho utilizada como exemplo mostraram que a amplitude da depressão por endogamia na geração S¥ de autofecundação é de 48,7% a 85,3% para produção de grãos e de 13,9% a 24,5% para altura da planta. Um delineamento apropriado para estimar a variância da depressão por endogamia, a sua amplitude e a amplitude de linhagens endogâmicas é apresentado.

REFERENCES

Benson, D.L. and Hallauer, A.R. (1994). Inbreeding depression rates in maize populations before and after recurrent selection. J. Hered. 85: 122-128.

Cockerham, C.C. (1983). Covariances of relatives from self-fertilization. Crop Sci. 23: 1177-1180.

Cockerham, C.C. and Matzinger, D.F. (1985). Selection response based on selfed progenies. Crop Sci. 25: 483-488.

Falconer, D.S. (1989). Introduction to Quantitative Genetics. 3rd edn. John Wiley and Sons, New York, pp. 438.

Gardner, C.O. (1965). Teoria de genética estadistica aplicable a las medias de variedades, sus cruces y poblaciones afines. Fitotec. Latinoam. 2: 11-22.

Gardner, C.O. and Lonnquist, J.H. (1959). Linkage and the degree of dominance of genes controlling quantitative characters in maize. Agron. J. 45: 186-191.

Hallauer, A.R. (1990). Methods used in developing maize inbreds. Maydica 35: 1-16.

Hallauer, A.R. and Miranda, J.B. (1988). Quantitative Genetics in Maize Breeding. Iowa State University Press, Ames, pp. 468.

Hallauer, A.R., Russell, W.A. and Smith, O.S. (1983). Quantitative analysis of Iowa Stiff Stalk Synthetic. Stadler Genet. Symp. 15: 83-104.

Lima, M., Miranda, J.B. and Gallo, P.B. (1984). Inbreeding depression in Brazilian populations of maize. Maydica 29: 203-215.

Miller, J.F. (1987). Sunflower. In: Principles of Cultivar Development (Fehr, W.R., ed.). Vol. 2. Macmillan Publishing Company, New York, pp. 626-668.

San Vicente, F.M. and Hallauer, A.R. (1993). Inbreeding depression rates of materials derived from two groups of maize inbred lines. Braz. J. Genet. 16: 989-1001.

Smith, G.A. (1987). Sugar beet. In: Principles of Cultivar Development (Fehr, W.R., ed.). Vol. 2. Macmillan Publishing Company, New York, pp. 577-625.

Souza Jr., C.L. (1985). Procedures for estimating expected genetic progress in inbred lines via recurrent intrapopulation selection. Braz. J. Genet. 8: 329-342.

Souza Jr., C.L. (1987). Reciprocal recurrent selection with half-sib progenies obtained alternately from non-inbred (S0) and inbred (S1) plants in maize. Maydica 32: 19-31.

Souza Jr., C.L. (1993). Comparisons of intra-, interpopulation, and modified recurrent selection methods. Braz. J. Genet. 16: 91-105.

Vianna, R.T., Gama, E.E.G., Naspolini Filho, V., Moro, J.R. and Vencovsky, R. (1982). Inbreeding depression of several introduced populations of maize. Maydica 27: 151-157.

(Received June 26, 1995)

Publication Dates

• Publication in this collection
  13 Oct 1998
• Date of issue
  Mar 1997

History

• Received
  26 June 1995