5:0“ mm ‘=" 335-331 ’31::‘311 33‘; 3F:L—ix?3d33~»1“1‘3 mag»; ;:::;':‘;':;1f‘311 it? CzijniL’LEFt; TRAETS 3. n «1 '1 n. -. f“ ,p a [‘- {‘3' 3” ? {L-‘Efi'r‘i ‘3‘ fl: 3 :2! ' Jun: "‘n’ I 5' '3‘ ‘ .'7. ‘ mm 2-3?- am war-war.“ of .42, S 3' ‘5r13 our 5 1'1! "3‘- ‘I. ‘3‘.“ ‘.- '1' ‘ ,. 171 ‘1 :Q’q: -.-.« - -*.~~' Iii: “1t!" a‘v" I‘irb fit 15‘ {5“ 4.? P‘ .- “. u: “é! PM“? 3 . ’ . ‘ F :93» E‘ «‘fi5147fl'f3 ”v: .. 3:: .‘g the ‘é‘ "1;. ‘3‘! .‘4: w! 31.- V? ii i? Shah ‘ LIBRARY Michigan State University COMPONENT INTERACTION IN RELATION TO MEAN EXPRESSION OF COMPLEX TRAITS IN A FIELD BEAN CROSS BY Rodrigo A. Duarte AN ABSTRACT Submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Farm Crops Year 1961 Approved WW-fi/ Rodrigo A. Duarte ABSTRACT The complex characters Total Leaf Area (T), Seed Yield (W), and Seed Size (V) in field beans (Phaseolus vulgaris) were analyzed by partitioning them into simpler components. For the purpose of the discussion, a complex trait was defined as one that has a physiological and/or morphological component sxuctuna The population studied consisted of progeny obtained . from intra-specific crosses of Phaseolus vulgaris, variety Algarrobo, a kidney bean from Colombia, South America, by Michelite, a navy bean variety from Michigan. During 1960, 1' F2 and F3 generations were grown in the field as parental, F well as in the greenhouse. The product of Number of leaflets per plant (N), and the average Size of the leaflet (S) was Total Leaf Area (T); Seed yield (W) was partitioned into Number of pods per plant (X), Number of seeds per pod (Y), and weight of the seed (Z); Length (L), Width (Wi), and Depth (D) of the seed were pro- posed as components of the Size of the seed (V). Independent genetic systems for N and S, components of Total leaf area (T), were postulated when a non-significant correlation between them was found. A similar situation pre- vailed for the yield components, X,1L and Z. Contrary to these findings, highly significant correlations (positive) were obtained Rodrigo A. Duarte between the seed size components, L, Wi, and D, suggesting that seed size is an allometric trait with a single genetic basis rather than a complex trait depending on interactions of independent gene systems. Number of leaflets per plant (N), and Size of the leaflet (S) were found to be influenced by dominance and addi- tive genetic systems respectively, i.e. complete dominance for high number of leaflets, and lack of dominance for the size of the leaflet. This conclusion was reached after testing the F1 against the parents and mid-parent; it was later supported byifle estimates of the average degree of dominance a, calculated from the partition of the variances of F2 and means of F3 into genetic variance and its components. Complete dominance for high number of pods, X, and no dominance for Y and Z was obtained in regard to the yield components. These findings implied that X was governed by a nonadditive genetic system, and Y'and Z by additive ones. A high degree of heterosis was observed in the complex traits, total leaf area (T), and seed yield (W). Since none of the components of these characters exhibited heterosis (or overdominant gene action), the heterosis observed was ascribed to the multiplicative effect. of the gene systems of the components, interacting at the level of morphological 4 Rodrigo A. Duarte integration, rather than at the nuclear or cytoplasmic level, i.e. a case of "component epistasis," in which the effect of one component on the complex trait, is conditioned by the value of the other component (or components), and vice versa. Theoretically additive and nonadditive genetic systems could be fixed in true breeding forms; therefore heterosis due to epistatic effects could be fixable also, by fixing sepa— rately the genetic systems of the components. COMPONENT INTERACTION IN RELATION TO MEAN EXPRESSION OF COMPLEX TRAITS IN A FIELD BEAN CROSS BY Rodrigo A. Duarte A THESIS Submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Farm Crops Year 1961 ii ACKNOWLEDGMENTS The author wishes to express his immense gratitude to Dr. M. W. Adams, for his guidance, criticism and help during the course of this study, and in the preparation of the manuscript. The author is very grateful to his wife Ligia, and his children for their inspiration, encouragement, and patience throughout the course of the studies. Finally, the author expresses his thanks to The Rockefeller Foundation and to the Departamento de Investi— gaciones Agropecuarias of Colombia, for financial assistance. TABLE OF CONTENTS INTRODUCTION . . REVIEW OF LITERATURE MATERIALS AND METHODS iii EXPERIMENTAL RESULTS AND DISCUSSION. 10 SUMMARY AND CONCLUSIONS LITERATURE CITED . 29 . 32 Table 10. ll. 12. LIST OF TABLES Correlation coefficients (r) measuring the inter- relationship of number of leaflets per plant (N), size of the leaflet (S), and total leaf area (T) in the F2 generation . . . . . . . . . . . . . . t-tests, variances, and mean values of leaflet number (N) of parents, mid-parent, and F1 . . . Variances in F and variances of the means of F progeny, and tfie average degree of dominance a, for number of leaflets (N) . . . . . . . . . . . t-tests, variances, and mean values of leaflet size (S) of parents, mid—parent, and F1 . . . . Variances in F2 and variances of the means of F3 progeny, and the degree of dominance a, for size of the leaflets (S) . . . . . . . . . . . . . . t-tests, variances, and mean values of total leaf area (T) for parents, sum of the parents, F O O I O O O O O O O O O O O O O O O O O O and l Variances in F2, and variances of the means of F3 progeny, and the degree of dominance 5, for total leaf area (T) . . . . . . . . . . . . . . Correlation coefficients (r) measuring the interdelationships of number of pods per plant (X), number of seeds per pod (Y), weight of the seed (Z), and total yield per plant (W), in the F2 generation . . . . . . . . . . . . . . . . . t-test, variances, and mean values of number of pods per plant (X) of the parents and F1 . . . . t-test, variances and mean values of number of seeds per pod (Y) for parents, mid-parent and F1 t-test, variances, and mean values of weight of the seed (Z) for parents, mid-parent, and F1 . . t'test, variances, and mean values of seed yield per plant (W) for parents and F1 . . . . . . . . iv Page 11 l3 l3 l6 l7 l9 19 21 23 24 24 25 Table Page 13. Correlation coefficients (r) measuring the inter— relationships between length (L), width (Wi), and depth (D), of the seed in the F2 generation . . . 27 INTRODUCTION Most phenotypic characters in plants are joint results of the actions and interactions of genes, together with envi— ronmental forces, that act upon the developing individual as it successively directs its transformation from the onset of germination to senescence. Certain traits are themselves nothing more than abstractions or artifacts compounded of two or more subsidiary or component traits. Yield of grain, for- age, and fiber in various plant species can be so categorized, that is, these are complex traits with a component structure. When the components are uncorrelated it is postulated that real genetic systems exist only for the components, and that the complex "trait" is only an interaction product of the components. To gain a fuller knowledge of the genetic behavior of major complex traits, plant breeders have found it expedient to partition the traits into simpler components for individual study. In this thesis the complex traits total leaf area, seed yield, and seed size in a field bean cross are subdivided into appropriate components for independent analysis. Total leaf area (T) is a complex trait independently of its rela- tbanship to other characters and is the product of number of leaves per plant (N) by mean leaf size (S). Components of yield (W) are number of pods per plant (X), average number of seeds per pod (Y), and average weight per seed (Z). The product of mean length of a seed (L), mean width (Wi), and mean depth (D) results in seed size (V), which in turn is one of the yield components. The main object of the study was to learn whether certain important traits in field beans have a component structure and whether independent analysis of the components would lead to any better understanding of the genetic basis of variation of the complex traits themselves. REVIEW OF LITERATURE Since the early years of the present century, when the first theories on the nature of heterosis were devised, the concept of heterosis due to interaction of the components of complex traits, which is essentially a developmental con- cept, has been over-shadowed by the more strictly genetic hypotheses of dominance and overdominance. More recently there has been a renewal of interest in the developmental concept in relation to heterosis for complex traits. Waddington (13) has proposed that genetic studies of phenotypic characters such as body weight or milk yield, should be done on bases of analysis of independent factors, that is,to try to partition the physiological system into less complex parts. He suggested that some genes might affect the milk yield by increasing the quantity of secreting tissue, others by affecting the efficiency of secretion, and perhaps some genes in still other ways. He also pointed out a case of variation in the quantity of vein formed in a region of Drosophila wing, in which the genetic systems fell into dis- tinct physiological groups in this way. Working with intra-specific crosses in Phaseolus vulgaris Sax (11) found a very close association between size and pigmentation of the seeds. It was reported that size ( differences even in the case of no dominance where several factors are involved, may be affected by the independent iaction of the size factors, which when combined have cumula- tive effects. Genotypic—environmental interactions influencing seed size in lima beans have been studied by Parsons and Allard (9). Seed size in lima beans was described as a complex 'trait made up of complicated interactions between genotype and micro-environment, and it was emphasized that seed size is one of the components of fitness in lima beans. According to Williams and Gilbert (16) yield heterosis in tomato hybrids could be explained by means of component interactions. Instances were described of yield heterosis in crosses when the components of yield were not heterotic. It was suggested as erroneous to speak of heterotic genes for complex traits such as crop yield. It was also mentioned that near maximal levels have been fixed in pure breeding varieties which fell in the upper ranges of variation and these were not exceeded by heterotic hybrids between poorer parents. Powers (10) reported a case of heterosis in yield of ripe fruit in tomato hybrids, due to intra— and inter-allelic interactions between components of the main trait, namely number of fruits that ripen, and weight per fruit. In turn number of fruits that ripen was found to be dependent on earliness of maturity, i.e. number of days from seeding to the first ripe fruit. Weight per fruit was partitioned into weight per locule and number of locules per fruit. Different degrees of dominance and heterosis were reported in the com— ponents as well as in the main characters. It was suggested that the study of genetics of heterosis could be simplified and improved by breaking the main traits down into their com- ponent characters. Working with cotton, Hutchinson (6) was able to analyze its yield components. They were: bolls per plant, seed cotton per boll, seeds per boll, lint per seed etc. Environmental variations seemed to affect more greatly some characters than others, and selection also was found to be more effective in certain characters. Perhaps the main idea of this work was shown in the compensatory variation of the components; that is, the intensification of one character can only be obtained at the expense of the others because of physiological incompa— tibilities. In a study of the breeding of self—pollinating cereals Whitehouse §§_§l, (15) reported yield components of wheat, using as the components: weight per grain, grains per spikelet, spikelets per ear, and ears per plant. Correlation analysis between components were made, and it was found they were com- pletely independent of each other. Yield predictions by means of diallel crosses, in which the best varieties for yield com— ponents were chosen, was also mentioned. 6 Grafius (3) has interpreted yield in oats as the volume of a rectangular parallelepiped, whose edges are the yield components: the number of panicles per unit area X, the average number of kernels per panicle Y, and the average kernel weight Z. It was pointed out that the edge most subject to change would be the longest and that changes in the components or edges would tend to counterbalance. In a study of heterosis in grain yield of barley, the same author (4) has clearly demonstrated the efficacy of the 3—dimensional component model. Components of yield in barley were the same as in oats except for X which in the case of barley was number of heads per plant. It was pointed out that yield is an artifact composed of "epistatic" interactions between components. Since X showed non-additive variability, further studies of this com- ponent were made, and it was found that earliness, or time, was the factor contributing to the dominance variability of heads per plant. In recent months, Grafius (5) has proposed ear number per plant (R), kernels per row (S), rows per ear (T), and kernel weight (U) as yield components in corn, assuming uni- form stahd. Frey (2) attributed yield in oats to the multiplicative interaction of components; because of this, the variety x location interaction variance was smaller for yield components than for grain yield. A new term "geometric epistasis, " was also proposed to denote the combination of components to produce grain yield. Yield of oat parents and progeny of crosses was discussed by Luedders (7) , using the yield components method. The results were quite similar to those Grafius (4) obtained in barley. MATERIALS AND METHODS Intra—specific crosses of Phaseolus vulgaris variety Algarrobo by variety Michelite were produced. Algarrobo is a mottled kidney bean variety from Colombia, South America, which possesses the determinate type of growth (bush type). Michelite is a navy bean variety, produced by the Michigan Agricultural Experiment Station. It has the indeterminate type of growth (vine type). Crosses were made under greenhouse conditions in 1959- 60, and F and F progenies were grown in that environment, 1 2 in order to give rise to F2 and F3 generations for planting in the field. During the summer of 1960, 25 F plants, 90 F 1 2 plants, 100 F3 families with 10 plants per each family, along with 20 plants for each one of the parents, were planted on land of the Michigan Agricultural Experiment Station, at East Lansing. Unfortunately bacterial and virus diseases decimated the population. Field data were obtained from individuals free of diseases, which consisted of 10 F plants, 24 F l 2 plants, 65 F3 families ranging from 2 to 5 plants in each family, and 10 plants for each of the parents. During the fall of 1960, from crosses made in the field during the summer, and F2 and F3 seeds harvested in the same field, a new population was grown in the greenhouse. Fifteen Fl plants, 200 F2 plants, 24 F3 families with 20 Iplants per family, in addition to 20 plants of each of the Ioarents,were grown. Because of the appearance of virus symp- tuoms on some plants, affected ones were eliminated. Data xvere taken from the following population: 6 F plants, 165 F l 2 Iplants, 23 families ranging from 2 to 17 plants in each fam- ily. Nineteen plants of Algarrobo, and 18 of Michelite were retained. Data on total leaf area and its components come from the field as well as from the greenhouse experiments. Yield and.seed size components data come from the field experiment only. Leaf area measurements and leaf counts were made two weeks after the onset of flowering. At maturity, number of pods per plant, and number of seeds per pod were recorded. In order to obtain the average weight of a seed, a sample of 100 seeds was taken. The 3-dimensional measurements of the seed to obtain seed size components were made on a random sample of 10 seeds per plant. 10 EXPERIMENTAL RESULTS AND DISCUSSION The results are presented according to the following outline: I. leaf Area A. Field results. B. Greenhouse results. 1960 (a) Correlation coefficients (b) Number of leaflets per plant (N) (c) Size of the leaflet (S) (d) Total leaf area per plant (T) II. M A. Field results (a) Correlation coefficients (b) Number of pods per plant (X) (c) Number of seeds per pod (Y) (d) weight of the seed (Z) (e) Total yield per plant (W) III. Seed Size (a) Correlation coefficients 11 I, Leaf Area A--Fie1d Results. B--Greenhouse Results (a) Correlation Coefficients The correlation coefficients of the number of leaflets per plant (N), size of the leaflet (S), and total leaf area (T) are presented in Table 1. TABLE 1. Correlation coefficients (r) measuring the inter- relationship of number of leaflets per plant (N), size of the leaflet (S), and total leaf area (T) in the F generation 2 A--Field Comparison df r N vs S 22 -.3230 N vs T 22 +.7931** S vs T 22 +.7692** B-—Greenhouse N vs S 160 -.l872 N vs T 160 +.8820** S vs T 160 +.826l** **P < .01 Negative but non—significant correlations were found for N vs S in the field as well as in the greenhouse experi- ments. The relationship is not so strong but that each one of the components (N and S) can be supposed to be conditioned largely by its own genetic system; changes in one of the 12 components only slightly affect changes in the other. As was expected, N and T and S and T are highly significantly corre- lated (positive). Changes in N, S, or both should affect total leaf area. (b) Number of Leaflets (N) From the t-tests given in Table 2 it may be seen that the F1 differs significantly from the mid—parent (m), but not from Michelite (P2) for average number of leaflets (N); this suggests complete dominance of genes for the high leaflet num- ber of Michelite, although interactions of non-alleles could not be excluded as a conditional possibility. In Table 3 an estimate of the average degree of dom- inance 5, of number of leaflets (N) from F2 and means of F3 progeny is presented, along with total observed variances and estimates of genetic and environmental variances and their components. The total variation measured in a population can be partitioned into components, using suitable models which have been developed. According to Mather (8), among others, total variance of segregating populations can be divided into three components: first, non—heritable variation due to environ- mental agencies (E); second, additive genetic variance (D); third, nonadditive or dominance genetic variance (H). The two 13 TABLE 2. t-tests, variances and mean values of leaflet number (N) of parents, mid—parent, and F1 . - Observed “““““ t-test """"" Generations N . . variance Comparison df t-value A—-Fie1d (+) P1 83.0 245.60 P . . . P . 2 320 0 1272 35 F1 vs 2 18 83 m 201.5 215.60 Fl 334.3 1682.81 F1 vs. m 18 9.63** B--Greenhouse P1 26.7 22.15 . . . P . P2 63 7 83 27 F1 vs 2 22 l 07 m 45.2 76.59 Fl 68.2 84.77 Fl vs. m 22 8.76** (+) For this and for the following tables P Michelite, m = Mid—parent. ** p < .01 = Algarrobo, P = 1 2 TABLE 3. Variances in F2 and variances of the means of F3 progeny, and the average degree of dominance 5, for number of leaflets (N) Gener- Observed Genetic components Of Variance a = ations variance variance H D E H/D A--Fie1d F 7347.17 6588.19 758.98 _2 9680.75 8306.01 1.08 F3 5012.00 4743.05 268.95 B--Greenhouse F 226.27 173.56 52.71 -2 135.73 279.25 .697 F3 153.29 148.11 5.18 l4 latter are heritable components of the variance. The variance of an F2 is VF2 = 1/2 D + 1/4 H + E1, and the variance of means of F3 families, VF3 = 1/2 D + 1/16 H + E2. As an example using the field results presented in Table 3, the total variance was partitioned as follows: VF = 1/2 D + 1/4 H + E 2 = 7,347.17 1 where environmental variation (E1) is measured by the mean variance of the parents (Table 2), VP +vr> ___1___Z. _ El 2 758.98. ‘nne~m”"-"“"fiw Genetic variance, GVFZ, is obtained by subtracting E1 from VF2: (1) GVF2 = 1/2 D + 1/4 H = 6,588.19. E2 for the variance of means of F3 families, VF3, is equal to 51 k I 0 ko being the adjusted mean of the number of individuals in each F3 family. The formula for finding k0, as given by Snedecor (12) . is 2 1 Zn ko q-l (Zn - Zn ) in which q represents the number of families in F and n the 3! number of individuals per family. For the present example k0 = 2.822. Therefore a Z§§42§-= 268.95. Genetic variance of the means of F ,GVF E2 2.822 3 3 is obtained by subtracting E2 from VF3. Therefore 15 (2) GVF3 = 1/2 D + 1/16 H = 4,743.05. Setting up the simultaneous equations (1) and (2), the values of H and D are found. (1) GVF2 1/2 D + 1/4 H = 6,588.19 (2) GVF3 1/2 D + 1/16 H = 4,743.05 :Subtracting equation (2) from equation (1): 3/16 H = 1,845.14 and H = 9,680.75. .Adding up equations (1) and (2): D + 5/16 H = 11,331.24 and D = 8,306.01 From the values of H and D it is possible to obtain, according to Mather (8), an estimate of the average level of dominance of the genes for a given character by means of the formula 5 = H/D, assuming no epistasis. Values of 5 near zero mean no dominance for either one of the parents, that is, addi— tivity for the genes concerned. Values close to unity suggest complete dominance for genes of one of the parents, really, nonadditivity of the genes. Values exceeding unity suggest overdominance. The value of 5 = 1.08 from the field results presented in Table 3, suggests complete dominance of the genes conditioning high number of leaflets and agrees completely with the results of the t—test between F1 and P2 (Table 2). Although the value of a = .697 from the greenhouse experiment is less than 1, it indicates a high degree of dominance, though not complete dominance. 16 (c) Leaflet Size (S) Table 4 shows comparisons of the F1 with the mid— parent (m) and with Algarrobo (Pl) for average size of the leaflets. The t—tests on field data imply no dominance for either one of the parents, that is, additivity of the genes that govern this trait, inasmuch as the F1 is not signifi- cantly different from the mid—parent (m). From the green- house results it appears that the F is significantly superior 1 to the mid-parent (m) at the 5 percent level. TABLE 4. t—tests, variances and mean values of leaflet size (S) of parents, mid-parent, and F1 S ---------- t—test ---------- Generations 2 Observed cms. variance Comparison df t-value A——Fie1d Pl 84.032 2.817 P2 33.161 1.864 m 58.596 1.664 Fl vs. m 18 1.38 F1 57.580 3.784 B-—Greenhouse Pl 147.16 593.47 Fl vs. P1 22 4.46** P2 63.94 134.88 m 105.55 280.10 Fl vs. m 22 2.47* Fl 118.37 62.27 *P < .05. **P < .01. 17 When the F1 is tested against Algarrobo (P1)' the parent with greater leaflet size, the test indicates signifi- cance at the 1 percent level for P1 over the F1’ suggesting no more than partial dominance for the greater leaflet size, even under artificial growing conditions. Estimates of the average degree of dominance 5 for the size of the leaflets (S) are presented in Table 5. Analysis of the field and greenhouse data results in values of 5 = .485 and 5 = .344 respectively, suggesting partial dominance for this particular component. However as the values of the addi- tive genetic variance (D) are compared with the nonadditive or dominant genetic variation (H), it is found that the former exceeds greatly the amount of the latter, indicating that the greatest portion of the genetic variance is due to additivity of the genes responsible for size of the leaflets. TABLE 5. Variances in F2, and variances of the means of F3 progeny, and the degree of dominance a, for size of the leaflets (S). Gener- Observed Genetic Components of Variance a = ations variance variance H D E H/D A—-Field F 274.237 271.998 2.239 _2 114.73 486.602 .485 F3 251.265 250.470 .795 B——Greenhouse F 852.98 488.80 364.18 _2 109.81 922.69 .344 F 503.98 468.21 35.77 18 (d) Total Leaf Area (T) Total leaf area (T), variances, mean values and t~test results are presented in Table 6. The F1 exceeds atiflmel percent level the better parent (P2) for total leaf area, and in fact- exceeds numerically the sum of both parents, in the field as inell as in the greenhouse, though not significantly. Heterosis for total leaf area (T) is postulated as a result of the multiplicative effect of the two components, one of them (N) under the control of genes with mostly dominant effects, and the other (S) under the control of genes largely additive in their action. Table 7 shows the average degree of dominance 5 = 3.066 and a = 2.850 for total leaf area (T) in the field and in the greenhouse, respectively. These values coincide completely with those presented in Table 6, which makes it appear that heterosis for this complex trait is due to overdominant loci in the F1. But since total leaf area is compounded of size (S) times number of leaflets (N), and these components do not exhibit heterosis (or overdominant gene action) in themselves, it is patently clear that the heterosis (and high level of overdominance) exhibited in the compounded trait, total leaf area, is due to the multiplicative effects inherent in the process of combining size and number of leaflets to get total leaf area. l9 I... TABLE 6. t—tests, variances, and mean values of total leaf area (T) for parents, sum of the parents, and F1 Gener- T Observed """"" t—test """"""" (ations cms. variance Comparison df t—value A--Field Pl 6,964.63 17,342,860 P2 10,611.59 870.422 Fl VS. P2 18 10.73** P +P . , . F . P +P . 1 2 17,586 22 1 265 107 1 vs 1 2 18 2 005 F1 19,249.19 5,604,327 B—-Greenhouse Pl 3,936.70 1,191,643 P2 4,109.17 1,146,336 Fl vs. P2 22 7.81** +P . , . F . P +P . 2 Pl 2 8,045 87 1 110 916 1 vs 1 2 22 01 F1 8,052.33 1,144.611 **P< .01 TABLE 7. Variances in F2, and variances of the means of F3 progeny, and the degree of dominance 5, for total leaf area (T) Gener- Observed Genetic Components of Variance 5 = ations variance variance H D E H/D A--Field F2 22,603,239 13,496,598 9,106,641 _ 44,516,674 4,734,338 3.066 F3 8,376,217 5,149,200 3,227.016 B——Greenhouse F2 3,870,248 2,701,258 1,168,989 8,661,197 1,070,903 2.850 F 1,191,100 1,076,268 114,832 20 Multiplicative interaction between leaflet number (N) ;and.1eaflet size (S) may be interpreted to mean that a condi- 'tion of epistasis exists for the complex trait. Epistasis in statistical-genetic language has been defined (Grafius, 4) as an interaction term containing variances due to interactions of additive x additive, additive x non- additive, nonadditive x nonadditive effects, and so on. According to Cockerham (1) epistasis in the simplest cases results from the joint action of two genes, one acting additively and the other dominantly, symbolized as V A-D. Since part of the variance due to dominant genes behaves in an additive manner, some variance of this kind also exists, sym- bolized by V A-A. The case of heterosis of total leaf area (T) fits the Cockerham model except that the interaction be- tween components in this case, is at the morphological level rather than at the intracellular level or, in other words, is a kind of somatic interaction. In Frey's (2) terminology, this is a case of "geometric epistasis" which denotes combination of components to produce total leaf area. 21 1;, Yield A—-l960 Field Results (a) Correlation Coefficients Correlation coefficients are given in Table 8 showing the relationships between number of pods per plant (X), num- ber of seeds per pod (Y), weight of the seed (Z), and total seed yield per plant (W), in the F generation. 2 TABLE 8. Correlation coefficients (r) measuring the inter— relationships of number of pods per plant (X), number of seeds per pod (Y), weight of the seed (Z), and total yield per plant (W), in the F2 generation. Comparison df r X vs. Y 22 +.0729 X vs. Z 22 -.0189 Y vs. Z 22 -.3954 X VS. W 22 +.8630** Y vs. W 22 +.8910** Z vs. W 22 +.8321** **P < .01 Non—significant correlations are found for X vs. Y, X vs. Z, and Y vs. Z, indicating that changes in one or two of the components do not greatly affect the remainder, that is, that variation in each component is largely independent of variation in the others. The negative association between Y and Z, however, is probably a real one, and not unexpected since develOpment would tend to impose a condition of 22 complementarity between number of seeds per pod (Y) and size of the seed (Z). The genetic relationship between Y and Z, ihoweVer, is not revealed by this association. This case is similar to that involving the components of leaf area as discussed above; in general, the values of the correlation coefficients imply that each component has for the most part an independent genetic system. Highly significant correlation coefficients are ob- served between each one of the yield components, X, Y, and Z, and the total seed yield per plant (W), as was expected. From these results, coupled with the fact that the components show independence each to the other, it is evident that the total seed yield per plant is due to the product of the three com- ponents, namely: number of pods per plant (X); number of seeds per pod (Y); and average weight of the seed (Z). (b) Number of Pods per Plant (X) Table 9 shows the t-test, variances, and mean values of number of pods per plant (X) of the parents and F1. Com- parison between F1 and Michelite (P21 the parent with higher number of pods per plant, indicates non-significant difference between them, suggesting complete dominance of the genes, that is, that a nonadditive genetic system is governing this character. Values of the degree of dominance a, and the com- ponents of variance in F and means of F3 families, are not 2 23