Analysis of Heterosis by a Direct Method Using the Concept of Heritability

Genetica114:163–170,2002.

?2002Kluwer Academic Publishers.Printed in the Netherlands.

163 Analysis of heterosis by a direct method using the concept of heritability

Zhong-xian(alias J.S.)Wu1&Ming D.Li2,?

1Laboratory of Quantitative Genetics,China Agricultural University,Beijing,The People’s Republic of China; 2Department of Pharmacology,College of Medicine,University of Tennessee Health Science Center,Mem-phis,TN38163USA;?Author for correspondence(Phone:(901)-448-6019;Fax:(901)-448-7300;E-mail: mdli@https://www.360docs.net/doc/2011280733.html,)

Received13September2001Accepted21December2001

Key words:crossheritability,dominance ratio,effective factors,heterosis,heterotic power

Abstract

The presence of heterosis has been observed in many species at both phenotypic and gene levels.Strangely,the genetic basis of heterosis was and still is largely unknown.In this study,we extended and simpli?ed some formulas that we reported previously.The foundation of our model was based on partitioning the F1phenotypic variance of the cross between two pure lines into additive,dominance and epistasis components,which lead to the estimation of effective factors,crossheritability in the broad and narrow sense and heterotic power.In the model,we assume that all polygenes controlling a quantitative trait have an equal genetic effect and are independent of each other.By extension of the heritability to a cross population,new features appear.The word‘crossheritability’acquires the status of a new genetic parameter that suf?ces to deal with the problem of crossbreeding and clari?es the picture of https://www.360docs.net/doc/2011280733.html,stly,an example of employing the proposed method in analyzing the crossing data from Drosophila melanogaster is given to illustrate its application.

Introduction

It has long been known that crossbreeding is an ef-?cient way to increase the productive ef?ciency of agricultural species(Duvick,2001).The major reason for this is the presence of heterosis(hybrid vigor).The phenomenon of heterosis has been recognized for al-most a century(Shull,1908)and exploited extensively in both animal and plant breeding over the years(e.g., Bell,1982;Visscher et al.,2000;Duvuck,2001).Het-erosis has also been reported at the molecular level. For example,a study of knockout mice heterozygous for a deletion of theα-CAMKII gene showed a signif-icant positive heterosis effect for defensive aggression (Chen et al.,1994).After reviewing phenotypic data in a number of species on more than a dozen of genes, Comings and MacMurray(2000)concluded that mo-lecular heterosis is common and may occur up to50% of all gene associations.Also,the same authors found that molecular heterosis may be gene-,phenotype-, and organ-speci?c.

Usually,two types of genetic variances of a breed/strain are of interest to animal and plant breed-ers,additive and non-additive genetic components.For the additive genetic fraction,a series of sophisticated methods to exploit them have been developed in past decades(e.g.,see Becker,1984;Falconer&Mackay, 1996).One of the most important concepts is herit-ability,which has been highly successful in the genetic analysis for pure breed population.On the other hand, the use of heritability in crossbreeding has been less fortunate and the genetic basis of heterosis is poorly understood(Wei et al.,1991;Duvick,2001).When many pure strains are crossed,the commonly used concept in heterosis analysis is called combining abil-ity;in which general combining ability is de?ned to be the average of all crosses while speci?c combining ability is the deviation of any cross from this average (Sprague&Tatum,1942;Mather&Jinks,1982).A number of models/methods has been proposed for es-timating heterosis based on additive,dominance and overdominance in plants(e.g.,Zhou,1982;Deng,

164

1998;Xu &Zhu,1999)and animals (e.g.,Dickerson,1973;Jakubec,1981;Hill,1982a;Wei et al.,1991).Additionally,several molecular approaches have been used to predict heterosis,such as molecular markers (Visscher et al.,2000)and DNA ?ngerprints (Gavora et al.,1996).

Relative to purebreding,crossbreeding has become more important today in terms of improvement of cur-rently available commercial species because almost all current commercial breeding animals have been highly selected for more than 50generations and most of the genes controlling economic traits may have been ?xed or toward to be ?xed through selection and inbreed-ing (Hill,1982b).Previously,we reported a heterosis analysis method for diploids under the simplest form of crossing (Wu &Li,1998).By introducing the concept of heritability into crossbreeding,we derived a series of formulas that can be used to estimate ef-fective factors controlling a trait,crossheritability in the broad and narrow sense,and heterotic power.In this study,we further extended and simpli?ed some of the methods reported previously (Wu &Li,1998).Moreover,we used a series of crossing data obtained from Drosophila melanogaster to illustrate the use of the proposed methods in heterosis analysis.Theoretical analysis

To start with,the usually accepted de?nition of het-erosis is (Dickerson,1969,1973;Sellier,1982)

H ij =F 1?MP ,

(1a)

where H ij is heterosis of the cross i ×j ,F 1is the phenotype of the ?rst ?lial generation and MP [=(P i +P j )/2]is the mid-parent value (see Figure 1).Working with second degree statistics,we ?nd the variance of heterosis to be

V H =V F 1+V MP ?2cov (F 1,MP ).It can be rewritten as

cov (F 1,MP )= 12

[V F 1+V MP ?V H ].

(1b)

Figure 1.Arbitrarily assigned genotypic values.The d repre-sents the deviation of A i A j from mid ?parent and a equals to

(P i ?P j )/2.

If we assume that the parent P i is homozygous for

most,if not all,of loci and the quantitative trait under consideration in P i is controlled by n i independent loci whose allele has an equal positive genetic ef-fect,then n i is a binomial variable with a probability of 0.5(Zhou,1982;Wu &Li,1988).Accordingly,E(n i )=N/2and V (n i )=N/4,where N is the total number of loci controlling the trait of interest and p =q =1/2in the F 1hybrids.The genotypic value of P i is (Wu &Li,1988)

P i =n i a +(N ?n i )(?a)+I i

=(2n i ?N)a +I i ,

E(P i )=E [(2n i ?N)a +I i ]=E(I),

where (N ?n i )is the number of loci with negative genetic effects on the trait and I i is epistasis effect in the i th parent population.If we take the mean as origin,then E(I)=0(see Figure 1).

Further,according to the formula of our previ-ous communication (Wu &Li,1988),we have the variance of P i as

V (P i )=V [2n i a +I i ]

=4a 2V (n i )+V I

=Na 2+V I .

Here,we assume there exists no covariance between additive and epistasis components.Then,

MP =

(P i +P j )

2

=(n i +n j ?N)a + 12

(I i +I j ),

V (MP )=V (n i +n j ?N)a + 12

(I i +I j )

= 12 Na 2+ 12 V I ,(2)where (1/2)Na 2is the additive variance in the F 1

generation and (1/2)V I the epistasis variance in the parental generation.Similarly,the number of hetero-zygous loci N ij in F ij is a multinomial variable with values 0,1,2,...,N .Unless N ij =N ,there will still exist homozygous loci of genes having ‘positive’or ‘negative’effects with respective numbers of

(n i +n j ?N ij )

2

for ‘+’effect genes ,

N ?(n i +n j ?N ij )2

for ‘?’effect genes .

Hence the genotypic value of F 1will be F 1=(n i +n j ?N)a +N ij d +I ij ,

E(F 1)= 12

Nd +E(I ij ),

V F 1= 12 Na 2+ 14 Nd 2

+V I (F 1),

(3)

165

where (1/4)Nd 2is the dominance variance and V I (F 1)the epistasis variance in the F 1generation,d =H =F 1?MP ,and we also assume that n i ,n j ,N ij and I ij are all independent of each other.The expression for heterosis now becomes

H ij =F 1?MP =N ij d +I ij ? 12

(I i +I j ),

V H = 14 Nd 2+V I (F 1)+ 12 V I .(4)Thus epistasis here has two components,parental [i.e.,(1/2)V I ]as well as F 1[i.e.,V I (F 1)].Solving Equations (2)–(4)for the genetic components,we have

12 Na 2= 12

[V F 1+V MP ?V H ],

(5) 12 Nd 2

+V I (F 1)

= 12 [V F 1+V H ?V MP ],(6) 12

V I =[V H +V MP ?V F 1],

(7)

where V MP ,V H and V F 1can be directly calculated from parental and F 1hybrid data.We see immediately that if we know V H ,V MP and V F 1,then (1/2)Na 2[i.e.,V A(F 1),additive variance]and V I can be easily ob-tained,while the non-additive variance [i.e.,V NA(F 1)]comes out in a lump together of (1/4)Nd 2and V I (F 1).Solving Equation (5)for N ,we have

N = 1

a 2 [V F 1+V MP ?V H ](8)

and the dominance ratio

d/a =2(F 1?MP )/(P i ?P j ).

(9)

Since V D(F 1)=(1/4)Nd 2(from Eq.(6)),we can ob-tain it by multiplication of (d/a)2and (1/2)Na 2and then dividing the product by 2,that is,

V D(F 1)=

[(d/a)2×V A(F 1)]

2

.(10)And from Equations (6)and (10),V I (F 1)can be easily obtained as

V I (F 1)=V NA(F 1)?V D(F 1).

(11)

Applying the principle of heritability in purebreeding to crossbreeding,we have

Crossheritability in the narrow sense

xh 2

=

12

Na 2V F 1

.(12)Crossheritability in the broad sense xH 2=

[(1/2)Na 2

+(1/4)Nd 2

+V I (F 1)]

V F 1

(13)

just as their corresponding counterparts are in pure-breeding,and Heterotic power :

H.P.=

V H

V F 1

=[(1/4)Nd 2+V I (F 1)+(1/2)V I ]V F 1

.

(14)

Materials and methods

Four stocks of Drosophila melanogaster were used and for convenience they are designated as 1,2,3and 4,respectively.The four stocks came originally from different localities in China and had been within-line inbred in the laboratory for at least 100generations before the start of the experiments.In addition they had been full-sib mated for 10more generations and showed no inbreeding depression for all traits con-sidered here.The traits of interest are body length,wing length,breast width,abdominal,bristle num-ber and body weight.Flies were maintained on corn medium at 25?C incubator.

The parental individuals of each cross were ran-domly chosen from these highly inbred stocks in an 1:1ratio of male to female and were virgin ?ies.Each mating was in a separate bottle.After mating the fer-tilized females were placed in single vials for 6days,one female ?y per vial.When eclosion began,virgin ?ies were taken out three times a day and the traits of interest measured immediately.During the whole stage of the experiments,there was no selection op-eration on any cross combination.The sample size of ?ies measured for the stocks varied within a range of 177–278.

The heritabilities for each trait of the parental stocks were computed using the full-sib method (Falconer &Mackay,1996),the means and standard deviations with the usual statistical techniques.The data for the estimation of the parameters of parental populations came from separate experiments designed speci?cally for that purpose,and is valid for the methods used in this report.Results

The phenotypic means and standard deviations of ?ve quantitative traits of interest for parental stocks 1–4are given in Table 1.Except for a few cross combinations

166

Table1.Phenotypic means,standard deviations,and heritabilities(in parenthesis)of parental lines

Stock Body length Wing length Breast width Bristle Body wt.

(dmm)(dmm)(dmm)number(mg)

169.18±2.8562.16±2.0825.31±0.8718.07±1.769.27±0.97

(0.189)(0.183)(0.163)(0.033)(0.249)

271.17±2.7164.60±2.1525.38±0.9417.22±1.7310.08±1.02

(0.401)(0.430)(0.143)0.354)(0.463)

369.47±2.7062.86±2.2228.83±0.8528.42±1.929.42±1.00

(0.247)(0.253)(0.300)(0.202)(0.413)

471.10±2.4563.43±2.0926.00±0.9717.77±1.6610.07±0.94

(0.163)(0.060)(0.159)(0.257)(0.313)

Table2.Means and standard deviations of the F1populations

Cross Sample size Body length Wing length Breast width Bristle Body wt.

(n)(dmm)(dmm)(dmm)number(mg)

1×216370.83±2.8563.54±2.3526.21±0.7918.02±1.5510.20±1.05

1×315469.83±2.0363.39±2.0426.17±0.6518.69±2.159.87±0.68

1×417970.39±2.4862.64±2.0226.10±0.7918.36±1.529.96±0.96

2×314571.20±2.2664.51±1.8226.46±0.8219.28±1.5010.25±0.82

2×416671.64±2.1864.66±1.9226.72±0.7717.89±1.5210.51±0.84

3×414970.83±2.0664.74±1.8524.65±0.6619.05±1.5610.24±0.75 for body length and body weight,most crosses have

signi?cant(p<0.05)or highly signi?cant(p<0.01)

differences between the means of the two crossed

stocks(data not shown).The estimated heritabilities

of four parental stocks for all?ve traits are also given

in Table1.It appears that all these traits had medium

or high heritabilities,and they were well within the

reported regions and are consistent with the expec-

tations from the nature of those traits.The means and

standard deviations of F1hybrids for all possible cross

combinations are shown in Table2.Only the crosses

were considered disregarding reciprocals and herein

we assumed that both the crosses and their reciprocals

have the same heterosis.

Using Equations(5)and(7)–(11)given above,

the parameters N,d/a,V A(F

1),V D(F

1)

,V I(F

1)

,and

V I,of only those crosses in which the differences between the crossed lines were statistically signi?cant, were estimated and are presented in Table3.While the additive variances((1/2)Na2,the3rd column of Table3)are relatively stable,results for N appear more sensitive since little changes in the relative size of a2lead to large de?ections in the values of N.Ex-cept for a couple of cases,such as crosses1×3and 1×4for bristle number,most of the estimated val-ues of genetic components including V A(F

1)

,V D(F

1)

, V I(F

1)

and V I are well compatible with one another.

The crossheritability in narrow sense and heterotic power of each cross are given in Table4and they were estimated from Equations(12)and(14),respectively. As predicted,the crosses with higher narrow cross-heritability(xh2)had lower heterotic power,indicat-ing that they will be unlikely to give good heterosis and vice versa.

Discussion

In this study,we reported a number of formulas that can be used to partition the phenotypic variance in the F1generation into the additive,dominance and epistasis components.From these variance compo-nents,we can obtain several genetic parameters such as the total number of effective factors controlling a quantitative trait(N),the dominance ratio(d/a), crossheritability in both broad(Xh2)and narrow sense (xh2),and heterotic power.With these parameters,it becomes possible to analyze a hybrid population in a more sophisticated way as we do for a purebreeding population.Thus,for any given hybrid population,as

167 Table3.Resolution of constituent terms of the F1populations

Cross Effective Dom.Ratio Add.Var.Dom.Var.F1Epi.Var.Par.Epi.Var.

factors(N)(d/a)V A(F

1)V D(F

1)

V I(F

1)

V I

Body length

1×27.62?0.66 3.760.82 3.550.20 1×47.32?0.27 3.310.12 2.670.37 2×39.01 1.03 3.26 1.740.120.78 3×48.79?0.67 2.900.660.680.83

Wing length

1×2 3.02?0.13 2.240.02 3.280.00 1×49.870.25 1.980.06 2.040.39 2×3 5.660.89 2.150.860.310.48 2×411.81 1.09 2.03 1.210.430.44

Breast width

1×2 2.71?0.680.390.090.140.03 1×30.220.510.340.040.040.06 1×47.11?1.300.420.35?0.150.01 2×30.540.940.400.180.100.00 2×419.24 2.710.36 1.33?1.100.18 3×40.36?1.950.360.68?0.600.11

Bristle number

1×216.540.89 1.480.580.340.25 1×3101.44?2.48 1.63 4.98?1.990.13 1×4126.73 2.99 1.36 6.09?5.140.20 2×38.97?2.42 1.63 4.74?4.110.26 2×436.71?1.41 1.40 1.39?0.480.25 3×429.59 2.91 1.57 6.66?5.800.09

Body weight

1×2 5.44?1.280.460.370.270.07 1×4 4.36?0.740.350.100.480.20 2×38.50 1.510.470.53?0.330.08 3×48.37?1.510.440.50?0.380.05

long as one knows crossheritability and selection in-tensity for the trait of interest,one would be able to predict the selection response from it.Additionally, with the concept of heterotic power,one would be able to determine whether a‘good’or‘bad’heterosis results from the cross of two pure lines.

The assumptions underlying our model are straightforward and simple.For the sake of simpli-city,we assume that all polygenes controlling a trait of interest have an equal effect and are independent of each other.We also assume that each parent is homo-zygous for almost all loci under the consideration.In some cases,the parental lines or breeds may not fully meet all assumptions underlying this model,but this should not prevent one from using the model proposed herein in their data analysis.Many methods,such as multivariate analysis(Li,1988),DNA?ngerprint (Gavora et al.,1996)and molecular markers(Visscher et al.,2000),have been proposed to predict heterosis, however,none of them has been proven to be supe-rior over others in terms of heterosis prediction for all cases.Therefore,using crossheritability and heterotic power to predict heterosis as proposed herein repre-sents an addition to the heterosis analysis methods in the literature.

Heterosis has been demonstrated in both plant and animal genetics and breeding.Any individual that was a hybrid of two strains likely showed higher yields than either parent lines(for a review,see Duvick, 2001).As a result,almost all economically plants and

168

Table4.Crossheritability and heterotic power of desired genetic parameters of cross population

Cross(h2i+h2j)/2Narrow Heterotic

crossheritability power

Body length

1×20.290.460.55

1×30.220.820.30

1×40.180.550.49

2×30.330.640.44

2×40.280.650.40

3×40.200.690.41

Wing length

1×20.310.410.60

1×30.220.380.80

1×40.120.490.56

2×30.340.650.42

2×40.250.650.51

3×40.160.650.39

Breast width

1×20.150.630.39

1×30.230.810.26

1×40.160.680.33

2×30.220.590.41

2×40.150.610.54

3×40.230.820.31

Bristle number

1×20.190.620.44

1×30.120.350.66

1×40.150.590.46

2×30.280.720.34

2×40.310.610.45

3×40.230.650.37

Body weight

1×20.360.420.62

1×30.330.600.85

1×40.280.380.73

2×30.440.700.35

2×40.390.580.52

3×40.360.780.27 animals now grown or raised throughout the world are hybrids.In addition to where hybrid vigor is observed at the individual level,signi?cant data have been ac-cumulated from recent molecular approach indicating that up to50%of genes show heterosis at molecular level as well(Comings&MacMurray,2000).This suggests that development of a comprehensive method of analyzing heterosis at the phenotypic,gene,gender and/or organ levels may have signi?cant implication in future research.

The heterosis is exactly as given by the formula of our theory except that it is divided by V F

1

to be comparable with the other components.It is clear that the additive fraction constitutes narrow crossherita-bility while the non-additive fraction heterotic power. Thus the genotype AaBbCc...in F1generation is transformed into meaningful genetic categories which can be evaluated in terms of known quantities N, a,d,or their allied combinative forms.Since the population comes into equilibrium after a single gen-eration of random mating,so that no further change in composition would occur from the F2generation and onwards.However,if selection is practiced in F2generation,there will be a genetic response based on the crossheritability,its size depending on the se-lection intensity,just as in purebreeding.Heterosis is seen here to consist of all heterozygous elements, that is,the non-additive components.No heterosis based on additivity is either present or possible,nor is there any need to invoke any other concept since heritability alone suf?ces to explain both pure-and cross-breeding.The reason is obvious:crossbreeding, in the sense of crossing between pure lines,where the gene frequency p=q=1/2,is a special case of purebreeding where p+q=1.

Classical genetic analysis of heterosis is con-cerned primarily with two items,gene number and dominance ratio.For the gene number,we accept the interpretation of Mather and Jinks as the num-ber of effective factors(Mather&Jinks,1982) and it is readily obtained from Equation(8)as (1/a2)[V F

1

+V MP?V H].Incidentally,this may rep-resent a simpler way of obtaining the number of genes controlling a trait than those previously proposed(e.g., see Mather&Jinks,1982;Falconer&Mackay,1996). Similarly,we will not linger over the question of the size of d/a.This is given by Equation(9)and comes out in double as‘+’and‘?’decimal values of the same?gure,as required by the theory.Overdominance (d/a>1)is rare,and we neglect it in our model although it occurs here in a few cases,for example, in bristle number and body weight.However,this does not imply that overdominance is not an impor-tant factor in the determination of heterosis.In some cases,overdominance may contribute signi?cantly to heterosis(Deng,1998;Deng,Fu&Lynch,1998).

As de?ned,crossheritability is heritability applied to a cross population.In this sense the pre?x‘cross’denotes that it is with crossbreeding we are concerned. As shown in Table4,when we compare the crossher-itability with its two separate heritabilities,it turns

169

out that its value is consistently higher.Thus with ex-tension of the concept to crossbreeding,new feature appears.This should not cause for wonder since with the crossing more genes and/or alleles per gene are introduced into the F1hybrids.This can only mean the birth of some new genetic parameter,crossheritability, with a range of0–1.Its place in crossbreeding should be equivalent to that of heritability in purebreed-ing.Since each cross has a separate crossheritabil-ity,crosses can be selected for just as traits in pure lines.With its installation,therefore,the twin?elds are uni?ed.Moreover,it has the merit that popula-tions with any foundation lines may be compared. However,this may not be the case with the combin-ing abilities.For example,general combining ability changes with the addition of each new line to the pop-ulation,and so does the speci?c combining ability change also(Sprague&Tatum,1942).This limitation alone shows that of the two,crossheritability may be preferred.

Crossheritability placed in proper perspective is rich in biological context apart from breeding.When two pure lines are crossed,we obtain a crossbred with a higher crossheritability;when two such cross-breds are crossed,we get a double crossbred with a still higher cross-crossheritability or,in shorter nota-tion,x2h2.If the process is inde?nitely continued,we obtain a crossbred of the n th order with x n h2approx-imating unity.Of course,some isolating mechanisms or new mutations must be assumed to get the change ?xed,but that will come in the course of time.There-fore,this may represent a link between quantitative genetics and evolution that provides a means of elevat-ing crossheritability in the cause of evolution–a?eld of evolution engineering that is worth exploring in the future.

Predicting heterosis is important and necessary. Several approaches such as DNA?ngerprint,molecu-lar marker and estimation of genetic distance between lines have been proposed.The presented method shows a door open to analyze heterosis in a relative simple fashion.There are limitations in the method reported here such as neglect of overdominance and linkage,which remains to be investigated in the future.

Acknowledgements

The authors are gratefully indebted to Professors Zhao-Bang Zeng(North Carolina State University) and Qin Zhang(China Agricultural University)for their valuable comments and suggestions on the manuscript.

References

Becker,W.A.,1984.Manual of Quantitative Genetics.Academic Enterprises,Pullman,4th edn.

Bell, A.E.,1982.Selection for heterosis–results with labora-tory and domestic animal.Proc.2nd World Congr.Genet.Appl.

Livest.Prod.6:206–227.

Chen,C.,D.G.Rainnie,R.W.Greene&S.Tonegawa,1994.Ab-normal fear response and aggressive behavior in mutant mice de?cient for alpha-calcium-calmodulin kinase II.Science266: 291–294.

Cockerham, C.C.,1954.An extension of the concept of par-titioning hereditary variance for analysis of covariances among relatives when epistasis is present.Genetics39:859–882.

Comings,D.E.&J.P.MacMurray,2000.Molecular heterosis:a review.Mol.Genet.Metab.71:19–31.

Deng,H.W.,1998.Estimating within-locus nonadditive coef?-cient and discriminating dominance versus overdominance as the genetic cause of heterosis.Genetics148:2003–2014.

Deng,H.W.,Y.X.Fu&M.Lynch,1998.Inferring the major genomic mode of dominance and overdominance.Genetica102–103:559–567.

Dickerson,G.E.,1973.Inbreeding and heterosis in animals,pp.54–77in Proc.Anim.Breed.Genet.Symp.In honor of J.L.Lush, Champaign,IL.

Dickerson,G.E.,1969.Experimental approaches in utilizing breed resources.Anim.Breed.Abstr.37:191–202.

Duvick,D.N.,2001.Biotechnology in the1930s:the development of hybrid maize.Nat.Rev.Genet.2:69–74.

Falconer,D.S.&T.F.C.Mackay,1996.Introduction to Quantitative Genetics.Longman,London,UK,4th edn.

Gavora,J.S.,R.W.Fairfull, B.F.Benkel,W.J.Cantwell&J.R.

Chambers,1996.Prediction of heterosis from DNA?ngerprints in chickens.Genetics144:777–784.

Hill,W.G.,1982a.Dominance and epistasis as components of heterosis.Z.Tierzuchtg.Zuchtungsbiol.99:161–168.

Hill,W.G.,1982b.Predictions of response to arti?cial selection from new mutations.Genet.Res.40:255–278.

Jakubec,V.,https://www.360docs.net/doc/2011280733.html,ponents of heterosis,pp.3.8–3.9in32nd Annual Meeting of the EAAP.Zegreb.

Li,M.D.,1988.Application of multivariate analysis to the predic-tion of animal heterosis.Acta Veterinaria et Zootechnica Sinica 19:1–5.

Kempthorne,O.,1957.An Introduction to Genetical Statistics.

Wiley,New York.

Mather,K.&J.L.Jinks,1982.Biometrical Genetics.University Press,Cambridge,3rd edn.

Sellier,P.,1982.Selection populations for use in crossbreed-ing.Proc.2nd World Congr.Genet.Appl.Livest.Prod.6: 15–49.

Shull,G.H.,1908.The composition of a?eld of maize.Am.Breed.

Assoc.Rep.4:296.

Sprague,G.F.&L.A.Tatum,1942.General versus special com-bining ability in single crosses of corn.J.Ame.Soc.Agron.34: 923–932.

Visscher,P.,R.Pong-Wong,C.Whittemore&C.Haley,2000.Im-pact of biotechnology on(cross)breeding programmes in pigs.

Livest.Prod.Sci.65:57–70.

170

Wei,M,H.A.M.Van der Steen,J.H.J.Van der Werf&W.E.

Brescamp,1991.Relationship between purebred and crossbred parameters I.Variances and covariances under the one-locus model.Z.Tierzuchtg.Zuchtungsbiol.108:253–261.

Wu,J.S.&M.D.Li,1988.An operational genetic theory of heterosis with applications in animal and plant breeding pp.

416–422in2nd Int.Conf.Quant.Genet.edited by B.S.Weir,

E.J.Eisen,M.M.Goodman&G.Namkoong.Sinauer Associates

Inc.,Sunderland,Massachusetts.

Xu,Z.C.&J.Zhu,1999.An approach for predicting heterosis based on an additive,dominance and additive×additive model with environment interaction.Heredity82:510–517.

Zhou,D.B.,1982.Approach to a mathematical model of heterosis of self-pollinated plants.Acta Genetica Sinica9:289–297.