Design I 11 With Marker Loci

Design I11 is an experimental design originally proposed by R. E. COMSTOCK and H. F. ROBINSON for estimating genetic variances and the average degree of dominance for quantitative trait loci (QTL) and has recently been extended for mapping QTL. In this paper, we first extend COMSTOCK and ROBINSON’S analysis of variance to include linkage, two-locus epistasis and the use of F3 parents. Then we develop the theory and statistical analysis of orthogonal contrasts and contrast X environment interaction for a single marker locus to characterize the effects of QTL. The methods are applied to the maize data of C. W. STUBER. The analyses strongly suggest that there are multiple linked QTL in many chromosomes for several traits examined. QTL effects are largely environment-independent for grain yield, ear height, plant height and ear leaf area and largely environment d pendent for days to tassel, grain moisture and ear number. There is significant QTL epistasis. The results are generally in favor of the hypothesis of dominance of favorable genes to explain the observed heterosis in grain yield and other traits, although epistasis could also play an important role and overdominance at individual QTL level can not be ruled out. D ESIGN I11 (experiment 111) was proposed by COMSTOCK and ROBINSON (1952) for the purpose of estimating the average degree of dominance for quantitative trait loci (QTL) . Random F2 individuals from a cross of two inbred lines were each backcrossed to both parental lines, and a quantitative trait was recorded for the progenies. An analysis of variance of the progenies provided estimates of quadratic functions of additive effects and of dominance effects of genes with the average degree of dominance being inferred from the ratio of the two. When the F2 individuals are genotyped for marker loci, this design provides some unusual features for estimating and testing for correlated (due to linkage disequilibrium) effects of linked QTL. Four orthogonal contrasts among the means of progenies for marker genotype X parental line combinations are used. Without epistasis, one contrast provides an estimate and test for additive effects of linked QTL, and another an estimate and test for dominance effects, whereas the other two have an expectation of zero. Epistatic effects of two or more QTL linked to the marker contribute to all four contrasts, with the contrasts that were zero without epistasis, providing distinct estimates and tests for epistasis. A modification of design I11 was used by STUBER et al. (1992) in that F3 individuals, each obtained by selffertilization of a distinct F2 parent, were backcrossed to the parental lines. Also, STUBER et al. determined the Cmesponding authur: C. Clark Cockerham, Program in Statistical Genetics, Department of Statistics, North Carolina State University, Raleigh, NC 276958203, E-mail: [email protected] genotypic constitution of the E; parents for several marker loci in order to detect and account for the variation due to linked QTL, which was the main purpose of the experiment. All of the analyses of STUBER et al. were conducted separately on the progenies from each parental line. The same analysis of variance proposed by COMSTOCK and ROBINSON and the same orthogonal contrasts can be applied to these progenies, requiring only a modification of the coefficients of the gene effects or in the quadratic functions of the gene effects in interpretation of the results. We briefly review the analysis of variance by COMSTOCK and ROBINSON and extend it to include linkage, two locus epistasis and the use of F 3 parents. We then develop the theory of orthogonal contrasts for a single marker followed by methods of statistical analysis. The test of genotype X environment interaction for the four orthogonal contrasts is also developed. The methods are then applied to the data of STUBER et al. ANALYSIS OF VARIANCE COMSTOCK and ROBINSON presented the analysis of variance for the progenies of the F2 backcrosses. We simply copy their Table 30.3 and present it as Table 1. They partitioned their progenies into sets that were replicated separately in order to reduce the environmental error. COMSTOCK and ROBINSON documented in detail the genetic constitution of a t and a$ along with the required assumptions and showed the effects of linkage. Since we want to extend the analysis to F3 parents, we outline the development. We consider a QTL with alGenetics 143 1437-1456 (July, 1996) 1438 C. C. Cockerham and Z.-B. Zeng TABLE 1 Analysis of variance from CoMsToCK and ROBINSON Source of variance d.f." Mean square Expected mean squareb Sets s 1 Replications in sets 4 p 1) Inbred lines in sets S F2 parents in sets s ( n 1) ms I u2 + 2mk f12 parents X lines in sets s ( n 1) %2 u2 + m:,, Remainder s(2n l ) ( p 1) %3 UZ " s, number of sets; n, number of F2 parents in a set; p, number of replicates. ' o:, progeny variance arising from difference among F2 parents; progeny variance arising from interaction o€k2 &d inbred parents. leles B and b. The genotypes of the inbred lines are denoted as BB for L2 and bb for Ll. The gene effects u, au and "u used by COMSTOCK and ROBINSON are changed to a d / 2 , d / 2 and -a d / 2 for genotypes BB, Bb and bb, respectively, for the purpose of expressing the effects as deviations from the mean (see APPENDIX A). Both provide the same variance. The effects are additive, a, and dominance, d. Genotypes of parents and progenies and means and differences of progenies for each parent in terms of gene effects are given in Table 2. The average of the means is a = Z: fi, and that of the differences is 3 = C fa, where the summation is over the parental genotypes. We find the variance among means as 02 = C fa' a' = a?/S for F2 parents and 3u2/16 for F3 parents. As pointed out by COMSTOCK and ROBINSON, the parent by line interaction variance is given by one half the variance of the differences. Consequently, ai/2 = [E fa' T 2 ] / 2 = d 2 / 4 for I$ parents and 3&/8 for F3 parents. COMSTOCK and ROE INSON summed these variances over independent loci (indexed by j here) so that a i = E] ~#/8 and aLl = Ej d;/4 for F2 parents. For F3 parents, a i = 3 Ej 4/16 and a:l = 3 X, d:/8. Expressing the degree of dominance as Dj = dj/ ai for the jth locus, the average squared degree of dominance is to be inferredfor F2 and F3 parents from the ratio a f J ( 2 a f ) = @ which is a weighted average with weights 4. COMSTOCK and ROE INSON used z (which is equivalent to Din our notations) to denote d m and called it the average degree of dominance. We extend the analysis to include linkage and twolocus epistasis. Let the gene effects for the second locus be dJ2, 4/2, -% d2/2 for genotypes CC, Cc, cc, respectively. Epistatic effects are denoted by E for additive X additive ( a X a ) , 4 for additive at B locus X dominance at C locus ( a X d), y for dominance at B locus X additive at C locus ( d X a ) , and w for dominance X dominance ( d X d). The recombination fraction between the two loci is denoted by r. The details are presented in APPENDIX A. We summarize the results in Table 3. Additive effects now include a X d effects, 4 and y , and dominance effects include an a X a effect, 6 , regardless of linkage. Note that E is a two-gene interaction, as is dominance, but for two nonalleles instead of alleles. Epistatic effects also contribute directly to both variances involving ( E + w)' in af and (4 + 7)' in ail, even with r = Their coefficients are complicated functions of r, but are small in comparison to the coefficients of the other effects. For K2 parents the term r( 1 r) + r(1 2r)' has a maximum of '/', when r = l/Q, but has almost the same values of when r = or The main consequence of linkage is on the variance of additive effects and of dominance effects when r < For pairs of linked loci without epistasis COMSTOCK and ROBINSON found that a i contains product terms (1 2 r ) a l e / 4 , which is positive if the genes are in coupling (both plus genes in the same parental lines and al and e have the same sign) and is negative if the genes are in repulsion (plus genes in different parental lines and al and % have different signs). In contrast, o f l contains the product term (1 2r)d1&/2 for which