additive genetic component [d] = rded

Heredity (1976), 36 (1), 31-40 EFFECT OF GENE DISPERSION ON ESTIMATES OF COMPONENTS OF GENERATION MEANS AND VARIANCES N. E. M. JAYASEKARA* and J. L. JINKS Department of Genetics, University of Birmingham, Birmingham BIS 2TT Received R.vii.75 SUMMARY The components of generation means that measure gene action and interaction at homozygous loci have expectations that depend on the degree of association or dispersion of alleles of like effect at different loci. With anything less than complete association, estimates of these components do not necessarily reflect the relative directions, magnitudes or kinds of gene action or interaction present. To illustrate these expected consequences, the F5, F5 and backcross generation of two contrasting crosses between pairs of inbred lines of.wicotiana rustica have been raised in which the same alleles are segregating at the same loci but for loci contributing to variation in final height the alleles of like effect are predominantly associated in one cross and predominantly dispersed in the other. The simultaneous analysis of the data from the two crosses show all the expected effects of the differences in the degree of association or dispersion. In the dispersion cross the effects of gene action at homozygous loci arc underestimated and the interaction between homozygous loci is not detected at all. The large directional dominance component that is common to both crosses is in fact no greater than the additive component once the deflating effect of dispersion is removed, thus ruling out the presence of overdominance. No effects of association or dispersion on the components of variation could be detected nor would any be expected unless there were differences in the predominant linkage phase between the two crosses. Large and predictable effects on estimates of the number of effective factors are, however, demonstrable. 1. INTRODUcTION JINics AND JoNEs (1958) showed that the additive genetic component and the homozygote x homozygote and homozygote x heterozygote non-allelic interaction components that can be estimated from generation means, have the following expectations: additive genetic component [d] = rded homozygote x homozygote ri kr 1 L5J interaction component k 1 homozygote x heterozygote - Lu rj interaction j. component All three components depend on some form of r, which measures the degree of association-dispersion of alleles of like effect in the parents, as much as on * Present address: Department of Genetics and Plant Breeding, Rubber Research Institute, Matugama, Sri Lanka. 31

32 N. E. M. JAYASEKARA AND J. L. JINKS the summed effects over the k loci of the appropriate gene action or interaction, the value of r ranging from 1 for complete association to 0 for complete dispersion (Mather and Jinks, 1971). Components which are solely dependent on gene action and interaction at heterozygous loci, for example, dominance [Jz], and heterozygote x heterozygote interaction [1], are independent of r. Hence in the absence of information on the value of r we can neither infer the level of dominance nor the level and type of interaction from the relative magnitudes of these components and when r approaches zero, i.e. complete dispersion, we may even fail to detect the contributions of gene action and interaction at homozygous loci. These predictable consequences of dispersion have not been demonstrated in practice, although frequently invoked to interpret the relative magnitudes of estimates of these components. Demonstration of these effects would require two pairs of true-breeding lines and generations derived from the two crosses between them in which the same alleles were segregating at the same loci but with quite different degrees of associationdispersion of alleles of like effect within the two pairs of parents. This requirement has now been met in inbred lines derived from a cross between varieties 1 and 5 of jvicotiana rustica. The data yielded by two sets of crosses which meet these criteria allow us to examine some other possible consequences of association-dispersion. For example, any linkages in the predominantly dispersion cross are more likely to be in the repulsion phase while in the predominantly association cross they are more likely to be in the coupling phase. Differences in linkage phase would in turn lead to characteristic differences in the magnitudes of the variances of corresponding segregating families from the two sets of crosses (Mather and Jinks, 1971). The expected deflationary effect of dispersion on estimates of the number of effective factors based upon comparisons of additive genetical components of means and variances can also be examined. 2. THE MATERIAL The allelic differences for final height, flowering time and related characters are largely dispersed between varieties I and 5 (Jinks and Perkins, 1969, 1972; Perkins and Jinks, 1973). Amongst the random sample of 82 inbred lines derived from the cross between 1 and 5 by Perkins and Jinks, two, B2 and B35 when grown in 1970, were the shortest and tallest at flowering time and among the earliest and latest to flower. While, therefore, the allelic differences between B2 and B35 for these characters are expected to be the same as those between 1 and 5 they must be predominantly in the associated phase. The degrees of association-dispersion (Td) for the two pairs of inbred lines, estimated as the proportion of the total range (2Ed) for any character covered by each pair of inbred lines (2r.d), are as follows (Jinks and Perkins, 1972). 1 and 5 B2 and B35 rdl r2 Final height 023 100 Flowering time 006 07l Height at flowering time 005 098

EFFECT OF GENE DISPERSION 33 For these three characters, therefore, the generations that can be derived from the I x 5 and the B2 x B35 crosses should allow us to estimate the same components of the generation means with widely different values of r but with the same gene action and interaction. To achieve this the F1, F2 and first backcross generations (B1 and B2) of the two crosses 1 x 5 and B2 x B35 have been raised with all possible reciprocal families. TABLE 1 The generation means, pooled over reciprocal and replicate families for the six generations of the two crosses i x and B2 x B35for the three characters final height, flowering time and height at flowering time. Apart from B35 where only 70 plants were raised, all the other means are based on 120 to 210 plants Cross Generation 1 x 5 B2 x B35 Final height mean mean P1 127340±20873 143835±18819 P, 103l70±l2727 92265±21211 F, 130235± 1'8053 126285± l 7430 F, 126495± l 5012 124 890± l 5207 B1 132295± l 6274 138605± 18762 B, 118 190±16972 111045±15746 Flowering time P1 77255±06525 83650±07012 P, 72l80±06061 69835±07608 F, 73675±0 5723 73825±05736 F, 76130±05787 76600±05260 B1 77 250±05866 80 160±06992 B, 73600±05990 73405±04914 Height at flowering time P1 75910± 19249 97380± 1 8725 P, 60830±09124 51530± 1 4062 F, 72755± 15230 74385± 13331 F, 73 680± 1 3458 74605± 1 3536 B, 78150±16364 87480±18858 B, 68 470± 1 2625 63060± 10543 To equalise the amount of information from the different generations the number of individuals raised in each generation was made proportional to its expected variance (Jinks and Perkins, 1969). For the 1 x 5 cross these expected variances were in fact based on the average observed variances from the many previous occasions on which the families had been grown. For the B2 and B35 cross the observed variances were available for the two inbred lines only. However, in the absence of linkage, directional dominance and genotype x environment interactions, the variances of the F1, F2, B and B2 generations are expected to be the same as those of the 1 x 5 cross and they were assumed to be the same for the purpose of designing the experiment. The extent to which the design succeeded can be seen from the uniformity of the standard errors of the generation means listed in table 1. The 1 x 5 cross and the B2 x B35 cross were each represented by a total of 1000 plants divided among the various generations and families in proportion to their expected variances. In addition, 10 plants of each of the 82 inbred lines were also grown making a total of 2820 plants in all. These were 36/1 C

34 N. E. M. JAYASEKARA AND J. L. JINKS divided equally between two replicate blocks, each family being equally divided between them, and all plants in each block were individually randomised at time of sowing. The experiment was grown in 1973. Of the many characters scored, only three, final height, flowering time and height at flowering will be considered here because they provide contrasting degrees of association-dispersion between the two crosses. The results for these characters are summarised in table 1. 3. MODEL FITTING TO GENERATION MEANS The three parameters [m], [h] and [1] which are independent of the degree of association-dispersion are expected to be identical in the two crosses, while the remaining parameters [d], {i] and [j], which depend on r are expected to differ between the two crosses. We shall proceed, therefore, on this expectation to find an adequate model of gene action and interaction for each character using the method of weighted least squares and determining the goodness of fit of each model by a x2 test (Mather and Jinks, 1971). This can be achieved by fitting models of increasing complexity until one is found which gives a non-significant x2 with all constituent parameters significant. The simplest model (model 1 in table 2) assumes that only additive and dominance gene action are present, giving four parameters, [m], [h], [d]1 for cross I x 5 and [d]2 for cross B2 x B35 for simultaneous fitting to the 12 generation means (six from each cross). TABLE 2 The parameters and their coefficients in the four models fitted to the generations means Cross 1 x5 B2xB35 Generation Model I common parameters \ r P1 P2 F1 F2 B1 B2 P1 P2 F1 F2 B1 B2 1111 m 1 1 1 1 1 1 1 1 O0144O0l Model 2 additional parameters [d]5 1 1 f 0 0 0 0 [d]2 0 0 0 0 0 0 1 1 0 0 [h] Model S 100 100 [i]1 0000001 1 100 000000 [ij2 additional parameters [i]2 0000001 [j], 0 0 0 0 0 0 0 0 0 0 [1] Model 4 additional parameters 100 1-000000 [i]1 1 1001-1- [j] 0 0 0 0 1 1-0 0 0 0 0 0 [i]2 0000001 [j]2 0 0 0 0 0 0 0 0 0 0 1- I [1] 00l1-1-Iool1-*+

EFFECT OF GENE DISPERSION 35 The next level of complexity is given by model 2 (table 2) with the two additional parameters [i]1 and [i]2 to account for non-allelic interactions between homozygous combinations in crosses 1 x 5 and B2 x B35, respectively. Model 3 (table 2) adds the three parameters [i]2, [jj2 and [1] to the four of model 1 primarily to test the prediction that a high degree of dispersion (r very small) will reduce any interactions specific to the 1 x 5 cross, that is [i]1 and [j]1, to non-significance. TABLE 3 The x' testing the goodness offit of the four models when fitted simultaneously to the 12 generation means of the two crosses combined for each of the three characters Model Character 1 2 3 4 Final height 1443 n.s. 523 no. 445 n.s. 344 n.s. Flowering time 35.65**** 594 n.s. 493 n.s. 225 n.s. Height at flowering time 21.58*** 286 n.s. 263 n.s. 134 n.s. Degrees of freedom 8 6 5 3 '' P<0001. ' " P = OOOl OOl. n.s. P>OO5. The final model (model 4 of table 2) adds all the parameters for nonallelic interactions to those of model 1 namely, [i]1, [i]2, [j]1, Eu 2 and [1]. The principal assumption being tested by this model is that m, [/à] and [1] are the same for both crosses and that only parameters dependent on r differ between them. This model would also fail, however, as would the simpler models if there was linkage among the interacting genes and the predominant phase of this linkage, i.e. coupling or repulsion, differed between the two crosses (Jinks and Perkins, 1969). The model fitting and testing procedures were carried out on the means of each of the 12 generations after pooling reciprocal families and replicate blocks. The x2 testing the goodness of fit of each of the four models to the three characters are listed in table 3. For final height the simplest model fitted is adequate and all parameters in this model (table 4) are significant. Attempts to simplify this model further by omitting parameters led to its failure. On adding interaction parameters, however, there is a marked and significant reduction in the x2 testing the goodness of fit but none of the additional parameters is itself significant. While we might suspect the presence of some low level of nonallelic interaction as previously reported for the 1 x 5 cross in some environments (Jinks, Perkins and Pooni, 1973) we have no evidence from the present analyses as to its nature. For flowering time the simplest model is inadequate but the inclusion of the two interaction parameters of model 2 or the three of model 3 gives an adequate fit that is not improved further by the additional interaction parameters of model 4. However, some of the interaction parameters in models 2 and 3 were not significant. These were removed in turn until an adequate model in which all parameters were significant was obtained. This proved to be the simplest model plus the interaction parameters [i]2 and [1] (table 4). The analysis of height at flowering time proceeded along the same lines and the final adequate model was again the simplest with the addition of and [1] (table 4).

36 N. E. M. JAYASEKARA AND J. L. JINKS TABLE 4 Estimates of the parameters in the simplest adequate model, in which every parameter is significant, for each of the three characters Parameter Estimate P Final height m 11768+08148 [d]1 13 37+ 1 0328 [d], 2630+l2183 "'"" [h] 12 70±15093 "'"" 1443 n.s. Flowering time m 7467+04350 'f,i'' [d], 277±03932 **** [d]2 6 81-I-0 4368 [h] 6 41±l 5632 ""' [i], 2 15±0 6436 **** [1] 733±14509 **** 46 920 n.s. Height at flowering time m 6869+10012 ',I'' [d]1 803±09405 **** [d], 2328+ 10160 **** {h] l4 32±35799 **** 5 83+14921 **** [1] 9l2±34l3l **** x6 299 For probability levels see table 3. The results of the model fitting confirm all our prior expectations and at the same time show that complex effects such as differences in linkage phase of interacting genes are not important in these data. For all three characters a common in, [h] and [1] is adequate for both crosses while different [d]s and [ijs are required. The [d]2 and [i]2 estimated from the association cross, B2 x B35 are significantly larger than the corresponding [d]1 and [i]1 estimated from the dispersion cross, 1 x5. Furthermore, the [i] type nonallelic interaction is significant and, therefore, detectable with some confidence only from the association cross, B2 x B35. In all of these respects the results follow the theoretical predictions. While the estimates of [d]2 are consistently greater than the corresponding estimates of {d]1 they are not as much greater as the estimates of the degrees of association-dispersion made before setting up the experiment led us to expect. For any one character the ratio of [d]2 to [d]1 should not differ from the ratio of the estimates of Td2 and r. But we find that the ratio of rds to r estimated in 1970 to be much the larger for all characters. For example, for final height rd2 is 43 times greater than rdl while [d]2 is only 20 times greater than [d]1. These two ratios, however, are based on different seasons, 1970 and 1973 respectively, and both ratios are sensitive to any differential changes in the relative performances of different genotypes over seasons. Such changes occur between 1970 and 1973 and are of a magnitude and direction to account for the apparent discrepancies. Using final height again as an example, in 1970, B2 and B35 were the extreme phenotypes for this character; in 1973, they covered only 81 per cent of the total range among

EFFECT OF GENE DISPERSION 37 the 82 inbred lines grown as controls. Varieties 1 and 5 in contrast had extended their range from 23 per cent to 38 per cent of the total. Therefore in respect of the genes controlling variation in final height in the 1973 season, B2 and B35 showed less association and varieties 1 and 5 more association than in 1970. 4. VARIANCES The estimates of the environmental component of variation (E1, the average within-family variances of the parental and F1 generations) the mean variance within the F2 families (V1p2) and the sum of the mean variances within the backcross families (V1B1+ V1B2) for each of the two crosses are listed in table 5. In the absence of genotype-environmental TABLE 5 Estimates of the mean variances within families for the two crosses averaged over reciprocal and replicate families Cross 1x5 B2xB35 Final height E1* 21386 22533 V1F2 26704 275l9 V1B,+ V1B, 49526 52478 Flowering time 24426 279O2 V1 4Ol95 33 060 V1BI+V1BZ 63105 624O5 Height at flowering time 15849 16294 V1F2 21736 218 95 V1B1+ V1B2 395 31 38978 * Estimated from the within family variances of the parental and F1 families as IVP1+Wp,+WF1. interactions and of linkage, these estimates should be the same for the two crosses providing that the same alleles are segregating at the same loci in both crosses. If linkage is present it will lead to differences between the estimates only if the predominant phase of linkage differs on balance between the two crosses. Furthermore, if the same alleles are not segregating at the same loci in both crosses it can only be because the derived inbred lines B2 and B35 carry the same allele at some of the loci at which the ancestral inbred lines I and 5 carried different alleles. Since a common m fits both crosses (table 4) any fixation must involve increasing alleles and decreasing alleles equally. In the present data these two causes of failure of the simple expectation will be opposing in action. Any differences in linkage phase are more likely to involve changes from predominantly repulsion linkages between 1 and 5, to predominantly coupling between B2 and B35 than the reverse and will, therefore, inflate the variances of the segregating generations of the B2 and B35 cross relative to those of the 1 x 5 cross. On the other hand, any chance fixation of the same alleles in B2 and B35 will deflate the variances of the

38 N. E. M. JAYASEKARA AND J. L. JINKS segregating generations of the B2 and B35 cross relative to those of the 1 x5 cross. The presence of genotype-environmental interactions could lead to differences between the estimates of E1 and of V1B1 + V1B2 for the two crosses but not to differences in V1p2 (Perkins and Jinks, 1970). Examination of the estimates of the corresponding variances for the two crosses show remarkably small differences if any between them for any of the three characters (table 5). The only difference of any note is that between the two estimates of V1p2 for flowering time that for the 1 x 5 cross being significantly the greater at the P = 0.04-0.06 level. Taken at its face value this result suggests that the same alleles have been fixed at some loci in the derived inbreds B2 and B35. As there is no supporting evidence from the backcross variances for flowering time or from any of the variances from the other two related characters it seems likely that sampling variation is responsible for this one border line significance out of the nine paired comparisons. From which we may conclude that there are no differences between the two crosses either in the incidence of genotype-environmental interactions or in linkage phase. The change from predominantly dispersion to predominantly association of the allelic differences has apparently affected neither. This does not mean, of course, that there is no genotype-environmental interactions or no linkage but only that if present their effects do not differ between the two crosses. There are insufficient statistics to test for linkage within either of the two crosses but we can test for genotype-environmental interactions for each cross separately by the standard method of comparing the variances of the parental and F1 families (Mather and Jinks, 1971). For final height and height at flowering time these variances are significantly heterogeneous for both crosses (4 final height, 1 x 5 cross 3847; B2 x B35 cross 2116; height at flowering time, 1 x 5 cross 7504; B2 x B35 cross 74.48). For flowering time, however, there is significant heterogeneity for the B2 x B35 cross only 1 (xi) x 5 cross 365; B2 x B35 cross 32.89). Hence only for this character has association led to greater sensitivity of the test for genotype-environmental interactions. Since epistasis is also present in these data (table 3) a simple additive, dominance and additive environmental model (D, H and E) of the family variances, which is the only one we can fit with the available statistics, is inadequate. While therefore, we can obtain perfect fit estimates of D, H and E they would be biased by the presence of these interactions. It is sufficient, however, for present purposes to know, as we have already established, that these components take the same values in the 1 x 5 and the B2 x B35 crosses. 5. NUMBER OF EFFECTIVE FACTORS An estimate of the number of effective factors, k, can be obtained as k [d]2 D This estimate assumes that the additive effects are the same at all gene loci and complete association (rd = 1) in which case [d]2 D /C2d2 lcd2 A:.

EFFECT OF GENE DISPERSION 39 If association is not complete (T< 1) k will be underestimated to the extent of r. Since D is the same for the 1 x 5 and the B2 x B35 crosses the relative magnitudes of It estimated for these two crosses are as [d], to [d]. Reference to table 4 shows that irrespective of the common value of D, estimates of the number of effective factors obtained from the B2 x B35 cross would be 387 times greater for final height, 604 times greater for flowering time and 840 times greater for height at flowering time than the corresponding estimates from the 1 x 5 cross. Since B2 x B35 is not the most completely associated cross possible for the gene differences between varieties 1 and 5, these large increases in the estimates of the number of effective factors obtained by using the B2 x B35 cross fall short of the maximum that could be achieved. 6. CoNcLusioNs Our analyses have illustrated all of the expected consequences of association-dispersion that are invoked to interpret the relative values of the components of generation means and estimates of the number of effective factors. For all three characters examined the apparent contribution of directional dominance is inflated in the dispersion cross while in the association cross the dominance component never exceeds the magnitude of the additive component. The results of the association cross thus remove overdominance as a possible explanation of the results of the dispersion cross and show that dispersion alone is responsible for the relatively high value of the dominance component. Since the same total number of individuals and the same types of families were raised for each cross, the two crosses can be compared directly for their efficiencies. From such comparisons the association cross emerges as the more efficient for detecting non-allelic interactions of the additive x additive kind. Association-dispersion per se has no effect on variance components. If, however the genes are linked, selection of an associated pair of lines such as B2 and B35 from a dispersion cross such as 1 x 5 could lead to differences in the predominant linkage phase between the association and dispersion cross and hence to differences in their variance components. No such differences could be detected in our data even though there is known to be repulsion linkages in the dispersion cross I x 5 (Perkins and Jinks, 1970). The test for differences in linkage phase, however, is relatively insensitive to all but the tighter linkages and there is little likelihood that these would have been recombined in the selfing programme that produced the associated pair of lines, B2 and B35. Acknowledgments. Financial support for N. E. M. Jayasekara was provided by the Technical Cooperative scheme of the Colombo Plan. 7. RItFERENCES JINKS, j. L., AND JoNEs, R. M. 1958. Estimation of the components of heterosis. Genetics, 43, 223-234. JINKS, j. L., AND PERKINS, j. M. 1969. The detection of linked epistatic genes for a metrical trait. Heredity, 24, 465-475.

40 N. E. M. JAYASEKARA AND J. L JINKS JINKS, j. L., AND PERKINS, p H. 1972. Predicting the range of inbred lines. Heredity, 28, 399-403. JINKS, J. L., PERKINS, 3. H., AND PooNs, H. s. 1973. The incidence of epistasis in normal and extreme environments. Heredity, 31, 263-269. s,saflser, K., AND JINKS, j. L. 1971. Biometrical Genetics, 2nd edition. Chapman and Hall, London. PERKINS, j. H.. AND JSNKS, j. L. 1970. Detection and estimation of genotype-environmental, linkage and epistatic components of variation for a metrical trait. Heredity, 25, 157-177. PERKINS, p H., AND finks, j. L. 1973. The assessment and specificity of environmental and genotype-environmental components of variability. Heredity, 30, 111-126.