Heredity 59 (1987) 105—115 The Genetical Society of Great Britain Received 2 October 1986 Expectation of means and variances of testcrosses produced from F2 and backcross individuals and their selfed progenies
A. E. Meichinger Institute of Plant Breeding, Seed Science and Population Genetics, University of Hohenheim, P.O. Box 70 05 62, D-7000 Stuttgart 70, F.R.G.
A biometrical genetic model is presented for the testcross performance of genotypes derived from a cross between two pure-breeding lines. The model is applied to obtain the genetical expectations of first andseconddegree statistics of testcrosses established from F2, first and higher backcross populations and their selfing generations. Theoretically, the testcross mean of these populations is expected to be a linear function of the percentage of germplasm from each parent line in the absence of epistasis. In both F2 and backcross populations, the new arising testcross variance between sublines is halved with each additional generation of selfing. Special consideration is given to the effects of linkage and epistasis, and tests for their presence are provided. The results are discussed with respect to implications in "second cycle" breeding. It is concluded that the choice of base populations between F2 and first backcrosses can be made on the distributions of testerosses from the first segregating generation. Schnell's (1983) "usefulness" criterion is recommended for choosing the optimum type of base population.
INTRODUCTION solution to the problem of choosing the most promising lines for improving the parents of a Inan advanced stage of hybrid breeding e.g., in single cross. Theoretical results for comparing F2 maize (Zea mays L.), new lines are predominantly and backcross populations with regard to per se developed either from advanced populations performance were provided by Mather, Jinks and undergoing recurrent selection or from ad hoc co-workers (see Mather and Jinks, 1982). However, synthesized populations between elitelines. no such theory is available with respect to the According to a survey by Bauman (1977), the testcross performance of these populations. majority of the most widely used public inbreds In this paper a biometrical genetic analysis is in commercial U.S. maize hybrids were "second presented for the testcross means and testcross cycle" inbreds of the latter type selected from F2 variances of F2 and arbitrary backcross popula- or backcross populations. Moreover, this author tions and their selfing generations developed from estimated that about 80 per cent of the effort in a cross between two pure-breeding lines. Special private maize breeding programmes is devoted to consideration is given to the effects of epistasis the improvement of established lines. and linkage and tests for their presence based on As in any breeding programme, the choice of first and second degree statistics are provided. In base materials is critical for success in "second addition, the results are discussed with regard to cycle" breeding. This holds true no matter whether implications for "second cycle" breeding. the pedigree or the single seed descent systems are used to extract new inbred lines with superior performance. In the case of "second cycle" breed- ASSUMPTIONSAND NOTATION ing, the choice of the base materials involves (a) choice of the parent lines and (b) a decision about Thetwo initial parents P1 and P2 of the popula- the type of base population to be established from tions subsequently considered are assumed to be these lines. In practice, F2 and backcrosses are homozygous lines. The F1 cross between P1 and most frequently used. P2 is backcrossed to each parent to produce the Concerning the first problem, Dudley (1984) Bi and B2 (also referred to as B11S0 and B21S0) recently presented a theoretical basis and possible generations, respectively. Backcrossing to parent 106 A. E. MELCHINGER line Pp, p =Ior 2, is continued by bulking pollen a superior T. Parameters dJ and d refer to half from a large (conceptionally infinite) number of the average effect of a gene substitution at the loci BPbSO plants producing generation Bph±ISO (b jandk in the gamete from the candidate, respec- 0). The population obtained from BphS() after n tively. Similarly, /krelatesto the additive by addi- generations of self-pollination is designated as tive epistatic effect between loci jandk. BpbS,,. For reasons of generality, the F1 is con- The above model is identical in form to the sidered as backcross generation zero and the F is model equation for the per se performance of referred to as Bp0S. diploid genotypes corresponding to the gamete of The tester T is assumed to be a homozygous the candidate (see Mather and Jinks, 1982). In line. In practice, the tester is often an elite line contrast to the parameters in equation (1), unrelated to P1 and P2 and with promising poten- however, those of the latter model are defined with tial as a parent in hybrid combinations with newly respect to the F., metric (Van der Veen, 1959). developed lines. The testcrosses are established by Despite this fundamental difference, the formal mating the candidates with the tester. analogy implies that part of the results given in In producing the backcrosses, selfed progenies, this paper correspond to previous results in and testcrosses, the following assumptions are literature concerning the per se performance of made: homozygous lines. Conversely, new results given (a)Normal diploid behaviour at meiosis here for the testcross performance apply directly (b)No gametic or zygotic selection to the per se performance of homozygous lines. (c)No mutation The above model equation also holds true for (d)Recombination frequencies are the same the expected testcross performance whatever the in the male and female gametogenesis and population structure of the tester may be (e.g., a independent of the cytoplasm and genetic single cross, synthetic or population), even if it is background. not in linkage equilibrium. However, heterogeneity In the genetic model, it is assumed that epistatic in the gametic array produced by the tester would interactions among three or more loci are absent. cause genetic variation within testcross progenies not considered here.
THEGENETIC MODEL LINKAGEDISEQUILIBRIUM Sincethe tester was assumed to be a homozygous line, differences in the genotypic values among Withregard to expressions for the means and testcross individuals are solely attributable to the variances given in subsequent sections, it is con- genotype of the gametes received from the test venient to provide general formulae describing the candidate. Consider two loci, jandk, with two linkage disequilibrium in the gametic arrays pro- alleles, A-a and B-b, present in parents P1 and duced by the populations investigated. P2. Employing the statistical model proposed for Using the conventional definition (Falconer, hybrid populations (Griffing, 1962; Schnell, 1965) 1981, p. 19), the linkage disequilibrium D3k for the present case, the genotypic values of test- between loci jandk in the gametic array produced crosses resulting from the four possible types of from cross P1 x P2 after n generations of selfing gametes produced by the candidate population can is given by the following expression (Cockerham be sub-divided in the following manner: and Weir, 1973): A,k TxAB= mT+dT±d+i] flD,k Af-_tk /tjk Tx Ab =mT+dT_dT_if,: x {1+(AJk/2)"(1— AJk)}Jk,(n 0); TxaB= mT_d+d_if,: (2) Txab=mT—dJ--d+if,:. (1) where Here the parameters on the right-hand side are for coupling linkage and statistically defined using as a base of reference for repulsion linkage in the gene-orthogonal population (Schnell, 1965) the parents, and between tester T and the F2-derived population AJk={1 of the cross P1 x P2 in linkage equilibrium. Their AJk denotes the linkage values between loci jand dependence on the tester genotype is indicated by k (Schnell, 1961). MEANS AND VARIANCES OF TESTCROSSES 107
In terms of the recombination frequency P1k then b,DJk =0for every b, n 0, i.e., the gametic betweenloci jandk, AJkisequal to I —2P3kand array of any backcross generation is in linkage hence becomes zero in the absence of linkage and equilibrium for unlinked loci. unity with complete linkage. Extending this result to backcrossing with sub- OVERALLMEANS OF TESTCROSSES sequent selfing, the linkage disequilibrium bflDfk Table 1 shows the frequencies of individual in the gametic array produced by generation BpbSfl genotypes in the bth backcross (BphSo) and those (p=l,2; b, nO) is obtained as obtained from each backcross genotype after n b,n'-jk =(1/4Y'1{(1+ tJk) (1 +4IflDjkI) —1}zJk. generations of selfing (BpbS) using results of (3) Jennings (1917). In addition, the testcross mean of Because of symmetry reasons, the linkage dis- BpbSQ- and BpbSfl-derived lines (adopting the ter- equilibrium is independent of p, i.e., identical for minology of Wricke and Weber, 1986, P. 73) is the two backcross series. Furthermore, if A3k =0 given for testcrosses produced from generation
Table 1. Genotype frequencies in populations BpbSo and BpbS, and genotypic testcross means of lines derived from them when testcrosses are produced in generation S, (t n)ona digenic interaction model with linkage; the recurrent backcross parent is assumed to have genotype AB/AB
Parent genotype in BpbSo AB/AB AB/Ab AB/aB AB/ab Testcross mean* of Bp5S-derived Frequency lines when testcr, are produced in S, 1 Genotype in CoeffIcient of Bp5S +Tbt 7b +Th df dT i] Frequency of descendants in Bp5S,, 2'—l 2—l 2'1 V"+Un AB/AB 1 — + § 1 1 1 2" 2'" 4 —1 1 V" AB/aB 0 0 1 0 0 2 2 —1 1 V" AB/aB 0 0 —--— 0 1 0 2 2"' 2 v + w" 0 0 0 0 AB/ab 2 ¶ 0 4,-nPjk 2"—1 2—1 V"—U Ab/Ab 0 0 + 1 —1 —1 2"" 2 v" -w' Ab/aB 0 0 0 0 0 2 ———1 V" Ab/ab 0 0 0 2"' 2 0 —1 0 2—1 2''—1 V"—U aB/aB 0 0 — + —1 1 —1 21 2+1 4 1 V" aB/ab 0 0 0 2"' 2 —1 0 0 2'—1 V'+U,, ab/ab 0 0 0 2"'+ 1
Testcr. mean* of BpbSo derived lines when tester, are prod. in S, dJ+d+i] itT d, 4,DJkitt * Genotypicdeviation from mT. 1' Tb{(1+AJk)1}/4. §V(1+Ak)/4; U={1—(AJk/2)}AJk/(2—AJk). ¶W=AJk/2. tt See equation (2). 108 A. E. MELCHINGER
S1(tn). Using these results, the following general populations are presented in table 2. For F1- formula is obtained for the expected mean of the derived generations, the results are analogous to testcrosses produced from population BpbS,: those for the per se performance of homozygous lines ______T p+i b T developed by dihaploidy or single seed des- Tx BpbS, =m+ (1){ 1 (1/2) }lI d I cent (Jinks and Pooni, 1981; Snape and Simpson, +{1 (1/2)b}2[jT] 1981).In the absence of epistasis, the testcross mean of the various populations is a linear function +4 Ib,(DjkIJkzfk, of [d T1, i.e., depends linearly on the percentage j Expected testcrosS mean __ epl'stasi'Sl % Germplasni 0 25 50 75 100 from P1 P2 B2 F= F7 51 P1 Generation Figure1 Graphical representation of the expected testcross means for various populations. The solid and the broken lines refer to the absence and presence of epistasis respectively. Calculations are based on dT =d=1and iJ assuming coupling phase in the parents and no linkage. MEANS AND VARIANCES OF TESTCROSSES 109 Table 2Expected testcross means of various populations on a digenic interaction model in the presence of linkage calculated from equation (3); for explanation of symbols see text Coefficient of Generation mT (_l)P+i[dT] [jT] 4fkjk TxPp 1 1 1 0 TxBp0S0=TxF1 1 0 0 A5 TXBp0S1=TxF2 1 0 0 AJk(1+AJk)/2 TXBp0S2=TxF3 1 0 0 AJk(4+Afk—AJk)/4(2—A,k) TxBp0S=TxF, 1 0 0 Afk/(2—Afk) 1 TXBp1S0=TxBp 1 4i Afk(2+AJk)/4 1 I TxBp1S1 1 4 )tJk(3+2AJk+Ak)/8 1 I TxBp1S,, 1 3AJS/4(2—AJk) 3 9 TxBp2S0 1 j- AJk(3+3AJk+AJk)/16 3 9 TxBp2S 1 j- AJk(5+2Afk)/16(2—AJk) For p =1,2. for situations of coupling linkage with duplicate +(_1)12{1 (1/2)'} epistasis or repulsion linkage with complementary epistasis, [jT] becomes negative. Since only the x[1 -{1 _(1/2)b}2) net effect can be observed, balancing positive and negative contributions may result in epistasis not being detected even when it is present. -4 (5) With net epistasis and no linkage among the b.ILkdJTiJ]. genes involved, equations (3) and (4) imply An analogous formula applies to the genetic vari- that the testcross means of higher selfing gener- ance for the per se performance of doubled ations do not differ from the testcross mean of the haploids derived from BpbS(. For b =0,i.e., gener- original backcross_generation_BpbSo. In particular, ations derived from the F1 , it agrees with the results the values for Tx F1, Tx F2, etc. and Tx Fe,. of Snape and Simpson (1981). should be identical (table 2). With linkage, Let gV(Pb; n) denote the total heritable test- however, these means may differ from each other. cross variance between BpbSfl-derived lines. This The size of the differences depends on the linkage variance is given by the summation values and is greatest for intermediate rather than extreme Ajk values. A comparison of Tx F1 and gV(Pb; n) = u2(pb; r); (6) Tx Fc,,. provides the most sensitive test for linked r=O epistatic effects. where o2(pb ;r) denotes the genetic testcross vari- ance between BpbSO-derived lines (r=0) or between BpbSr-derived sublines within Bp5Sri GENETICVARIANCES OF TESTCROSSES derived lines (r>0). Making use of the results in table 1, these Utilisingthe results in table 1 and the appropriate components of variance have the following general formula, the expectation of the total expectations (p =1,2; b 0): (between and within line) genetic variance among testcrosses produced from population BpbS, Model 1: No epistasis and no linkage becomes (p =1,2;b, tO): — u2(pb; 0) =(1/2)"{l(1/2)"}dj2; (7) Variance (Tx BpbSI) = [1—{l —(1/2)}2]dj2+4 b,fDjkdJdk r>0: o-2(pb; r) =(1/2)b+r (8) j +(1_{l_(1/2)b}4) Model 2: No epistasis but with linkage — 11 / /\b12o v' ri 2(pb; 0) = (1/2)b}dT2j 'Il1//.JJo L. b,tl-'jkljk (i/2)b[{i j 2 1 + A1k{:+AJk)/2}JkdJdT]; (10) j/k{(1+AJk)}2 0JJkdJiJ}. jk (14) Model3: With epistasis and no linkage Under models Ito 3 the formulae for a-2(ph; r) do not depend on the selfing generation St used — for producing the testcrosses. With both epistasis u2(pb; 0) =(1/2)b 1 (1/2)b}d2J i andlinkage, however, the testcross means of BPbSfl- derivedlines and consequently their variance may T2 /23b±i_5.22b+2b+2_l______ij change with additional selfing due to the oppor- 8 /J +(1['+(\ 2 [ A,k2—AJk )2 F2AND FIRST BACKCROSS DERIVED SELFING /4 GENERATIONS Relationships among genetic variances +(iYt' 2 _(1/2)b_i Letus now consider the results of the previous sectionin greater detail for the most important + 1_ specialcases h 0 and h =1,i.e., testcrosses estab- from F2 and first backcross populations and 2A;k)"((1/2)3'2— AJkAJk)} lished +(l their selfed progenies. For reasons of simplicity, := X (13) the abbreviations Po := 0 and Pip (p =1,2) are °jkdjujk}; subsequently used. With no epistasis and no linkage (model 1), thenew arising testcross variance between 02(pb; r) =formula(10) + (1/2)b+r sublines in F2 populations is halved with each further generation of selfing,i.e., the ratio Ij I Between F2-derived 0.2(0; 1) AJk/2 0 (2—2AJk+Ak—2Ak+Ak)/2(2—AJk)2 0 lines — Between F3-derived 0.2(0;2) (A3k/2)2 0 (1/2) —(1 + Ak)(3 Ak) 0 sublineswithin x (2— 2AJk + Ak)/ 8(2— F2-derived lines n + I Between F+2-derived 0.2(0; n + 1) (1/2Y1 (Ajk/2Y' (22 —3)/4" 0 (l/2) — {(1 + Ak)/4}{(3— Ak) 0 — sublines within x (2— 2AJk + )tfk)/2(2 A1j2} F1-derived lines Total var. 1 0 1 genetic 5V(0; ) AJk/(2—AJk) 1 —{A/(2—A)}2 0 between SSD lines developed from F2 — — — — Between Bp1S0- o-2(p; 0) * §1/4 (1 + AJk)(3 5Ak +2Ak)/4(2 §(1 Ak)/2(2 AJk) derived lines 2 Between Bp1S1- o-2(p; 1) (AJk/2)(1-1-AJk)/4 (1/2)—{(1+AJk)(3—Ak) — derived sublines x (2— 2AJk + AJk)/8(2 AJk)2} within Bp1S0- derived lines + — n I Between Bp1S-derived u2(p; n) (1/2)" (AJk/2)"(l+AJk)/4 (2+2_3)/4 (1/2) —{(1 + Ak)/4}''{(1+AJk)x AJk) sublines within (3— X) — Bp5S_1-derived lines x (2— 2AJk + A)/8(2 — — — — Total geneticvar. 5V(p; o) (3/4)AJk/(2 AJk) {15 6k k/(2 AJk) between SSD lines _9A%/(2_AJk)2}/16 (2—AJk) developed from Bp t Model 1: No epistasis and no linkage; model 2: no epistasis but with linkage; model 3: with epistasis and no linkage; model 4: with epistasis and linkage. §The +sign refers to p=l and the —sign to p=2. 112 A. E. MELCHINGER the absence of epistasis (model 2), the preponder- predictor is simpler and more rapidly obtainable ance of coupling or repulsion linkages enhances but is biased in the presence of epistasis. The or reduces this ratio, respectively. The quantitative second predictor is not biased by epistasis between effect depends on the linkage values. With unlinked loci and also yields good approximation extremely tight linkage, the ratio is hardly affected. in the presence of linkage. With extremely loose linkage, the contributions of Under model 2, the individual variance com- the cross product terms Jkdfd to 0.2(0; r)and ponents o-2(p; r) and consequently gV(p; cc) are gV(0; cc) are rather small. The greatest influence identical for both backcrosses (p =1,2) as pointed on deviations from the ratio of 2: 1 is exerted by out in the preceding section. Differences between gene pairs with intermediate to low linkage values, corresponding testcross variances in BI and B2 particularly during the first few generations of are attributable to the sum of cross product terms selfing. OjLJkdJzjk and therefore supply additional Epistasis between unlinked loci (model 3) leads evidence of epistasis. For dispersed gene associ- to an increase in the secondary (r= 2)and tertiary ation, OJkdJ Jk and OkLJkdiIkhaveopposing (r=3)variancesrelative to the primary (r= signs and hence their contributions cancel each 1)variance. In later selfing generations, epistasis other at least partly in the total sum. With coupling causes practically no deviation from the ratio2: 1. linkage, both products have the same sign. Linkage hardly affects the coefficients of i7,for Altogether, it can be concluded that the com- lower linkage values. With tighter linkage, the con- ponents of the testcross variance in Bi (the back- tributions of epistatic effects to 0.2(0;r) and cross to the higher performing parent with O =I gV(O; cc) decrease and finally vanish for com- at the majority of loci) are greater than those in pletely linked loci (AJk =1). B2 when there is predominantly complementary Under model 1, the genetic testcross variance epistasis between gene pairs in coupling. Con- between homozygous lines derived from F2, versely, the variances in Bi are smaller than in B2 gV(O; cc), is twice the testcross variance between with a preponderance of duplicate epistasis among F2 individuals, cr2 (0; 1). Hence, the latter can be loci in coupling. used to predict the former. With significant epis- Table 3 enables us to compare the testcross tasis, however, this predictor is biased. Consider- variances of F2 and backcross populations in corre- able bias can also be generated by linkage as was sponding generations. Under model 2, we have demonstrated by Kearsey (1985) for the per se performance of lines. u2(0; 1)=2cr2(p;0), (p=i,2). (15) The results for the backcrosses BI and B2 With epistasis among unlinked genes (model 3), (table 3) show that under model 1, o-2(p; 0), the we obtain testcross variance between Bp1S0-derived lines, is equal to a- 2(p; 1), the testcross variance between (72(1; 0)+2(2; 0) 0.2(0; 1) i2/8. sublines after the first selfing generation. The test- j Genotypic value P2 P1 Figure2 Assumed genotypic frequency distributions for the means of testcross progenies of random individuals from the F2 and backcross populations BI and 82inthe absence of epistasis. mT+[dT](i+l/V) and vice versa. We thus if the masking variances are relatively unimportant. obtain the remarkable result that the sign of The dependence of U(a) on aindicatesthat the 1r0(x)— ir1(x)depends solely on mT and [d ]je amount of resources allotted to a breeding pro- the testcross means of F2 and Bi but not on their gramme also influences the "usefulness". testcross variances. Due to this property, the prob- With "usefulness" as a criterion, the choice of abilities r(x) and ir1(x) provide only limited F2 vs. backcross base populations is reduced to information about the relative merits of F2 vs. the question whether the disadvantage of a lower backcross populations. population mean for the F2 testcrosses can be offset As an alternative criterion, I suggest employing by their greater selection response. This question Schnell's(1983) concept of "usefulness" cannot be answered in general but depends on the ("Brauchbarkeit") for assessing the breeding pros- specific conditions of the materials and trait(s) pects of base populations in "second cycle" breed- considered. In summary, it can be concluded that ing. He defined "usefulness" U(a) as the F2 is likely to have superior "usefulness" than B1 if U(a)=+R(a)=a+icrh (19) where M• denotes the population mean, R(a) = —the differences in the testcross means of the the F2 and Bi populations are small compared expected selection response when selecting the to the pertinent genetic standard deviations. upper a % phenotypes, er= thegenetic standard deviation, h =thesquare root of the heritability of —the heritability of the character is high, and —a high selection intensity can he applied. the considered trait, and,i,,the selection intensity when the upper a% phenotypes are In practice, the choice of F2 vs. backcross base selected. populations in "second cycle" breeding is compli- In other words, "usefulness" denotes the cated by the fact that the breeder regards not only expected genotypic mean of the upper a% a single trait but several characters simultaneously. phenotypes. Here, all quantities refer to testcross Despite this complication, the principles outlined performance. above should help in clarifying the various aspects Obviously, "usefulness" accounts for difleren- to be considered for this decision. Experimental ces in both means and genetic variances. Further- results related to the theory in this paper are in more, it demonstrates that the heritability should progress and will be published elsewhere. also be considered for assessing the prospects of obtaining and identifying superior genotypes. As follows from equation (15), h for the testcrosses Acknowledgements The author is indebted to Professor H. H. from F2 is about timesas great as h for those Geiger for his advice and helpful criticism in preparing the from BI if the masking variances (genotype by manuscript; to Drs H. C. Becker, U. Posselt and H. F. Utz for environment and error variances) are large, com- critical reading of the manuscript; to M. C. Dawson, M. A., for his correction of the English text; and to Mrs Schwarz and pared to the genetic variances. On the other hand, Mr W. I)aum for their assistance in typing the manuscript and h is of a similar size in both types of populations preparing the figures. MEANS AND VARIANCES OF TESTCROSSES 115 A simplified version of this paper was read at the sixth JINKS, .J.L.AND POoNI, H. 5.1981. Propertiesofpure-breeding meeting of the EUCARPIA Section "Biometrics in Plant Breed- lines produced by dihaploidy, single seed descent and ing" held in Birmingham, U.K., July 28-August 1, 1986. pedigree breeding. Heredity, 46, 391-395. JINKS, J. L. AND POONI, H. 5. 1984. Comparison of inbred lines produced by single seed descent and pedigree inbreeding. Heredity, 53, 299-308. REFERENCES KEARSEY, M. J. 1985. The effect of linkage on additive genetic variance with inbreeding an F2. Heredity, 55, 139—143. MATHER,K. AND JINKS, J.L.1982.Biometrical Genetics, 3rd BAUMAN,L. F. 1977. 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