THE QUANTITATIVE GENETICS OF HETEROSIS. Kendall R. Lamkey and Jode W. Edwards INTRODUCTION

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1 Lamkey and Edwards - THE QUANTITATIVE GENETICS OF HETEROSIS Kendall R. Lamkey and Jode W. Edwards INTRODUCTION Nearly 50 years have elased since the seminal heterosis conference was held at Iowa State College (Gowen, 95). That conference undoubtedly grew out of the obvious imortance of maize (Zea mays L.) hybrids in the agricultural economy of Iowa and the U.S. as well as the lack of understanding of the henomenon of heterosis. Farmers in Iowa raidly adoted maize hybrids. In just 5 years, Iowa went from 0 to 00% of the maize acreage being lanted to hybrids. Gowen (95) stated the following about hybrid maize It seems likely that in no other eriod of like years has there been such an increase in food roduced over so many acres of land. The return from hybrid corn has been henomenal, but it is now evidently aroaching an asymtotic value. If only Gowen could have looked ahead 50 years, because the best was yet to come (Fig. ). The term heterosis was coined by Shull (95). He defined the heterosis concet as the interretation of increased vigor, size, fruitfulness, seed of develoment, resistance to disease and to insect ests, or to climatic rigors of any kind, manifested by crossbred organisms as comared with corresonding inbreds, as the secific results of unlikeness in the constitutions of the uniting arental gametes. This definition is often interreted as not imlying a genetic basis for heterosis, because the definition basically describes the henotye that results from crossing two different inbred lines. For our uroses, we will define heterosis or hybrid vigor as the difference between the hybrid and the mean of the two arents (Falconer and Mackay, 996). This definition is usually called midarent heterosis. Midarent heterosis is often exressed as a ercentage of the midarent in the literature. It is imortant to note, however, that ercent midarent heterosis is difficult to interret from a quantitative genetic oint of view, and statistical tests of ercent midarent heterosis are nearly imossible. High arent heterosis is referred in some circumstances, articularly in self-ollinated cros, for which the goal is to find a better hybrid than either of the arents. To some, the terms hybrids and heterosis are synonymous. This is misleading, however, because there are hybrids that do not exhibit heterosis, but there cannot be heterosis without hybrids. In some secies, hybrids are sold commercially because crossing of two varieties brings together comlementary traits controlled by additive gene action. Distinguishing between hybrids and heterosis is imortant, because hybrids bring factors other than heterosis er se, i.e., uniformity, reroducibility, etc., to cro roduction. Often these factors are

2 Lamkey and Edwards - 0 U. S. Corn Yields and Cultivar Tyes 866 to Grain Yield (t/ha) Oen-ollinated b = Double-Crosses b = Single-Crosses b = Heritability Year Fig. Average U. S. maize yields from 866 to 996. Regression lines were slined together at 930 and 960, which corresonds roughly to when doublecross and single-cross hybrids started to become imortant. Regression coefficients are in t/ha - /year. Data are from the USDA, National Agricultural Statistics Service. confounded and difficult to searate. For examle, uniformity may result in higher yields. Is uniformity a genetic or nongenetic cause of increased yield? Is uniformity a factor in heterosis? In other secies (such as wheat), hybrids are being sought as a means to revent farmers from saving and lanting their own seed and as a means of rotecting research investments in transgenes. In this manuscrit we will review basic quantitative genetic concets in heterosis, introduce the concet of baseline heterosis, review the results of over 50 years of gene action studies, and suggest needs for future research. We feel that the quantitative genetics of heterosis must be tied to lant imrovement and that any theory of heterosis must exlain and be consistent with the increase in yield of hybrid maize since 930 (Fig. ). BASIC QUANTITATIVE GENETIC CONCEPTS OF HETEROSIS Much of the quantitative genetic theory of heterosis is based on single locus theory. Single locus heterosis theory assumes the absence of eistasis, which considerably simlifies the mathematics and interretations of the theory. Single locus theory will be reviewed in detail, because basic roerties of heterosis are derived from this theory. Willham and Pollak (985) develoed single locus heterosis theory for redicting the erformance of the F, F, arents, and the backcross to the arents. They used the random mated F (the F generation) as the base oulation in which all genetic effects are defined. Willham and Pollak (985) were interested in alying this theory to animals, for which inbreeding the arental oulations is rare. Therefore, we have extended this theory to include any level of inbreeding of the F, F, P, and P generations. A edigree showing how each

3 Lamkey and Edwards - 3 Po Po A + A A A RM X (P x P) F + RM Self + AA A A Inbred A A + A A A A + A A Random Mate (P x P) F + A A RM + A A A A Inbred Self Fig.. Mating scheme for oulations described in text. Poulations and start with the gametic arrays shown and are in random mating equilibrium (anmixia). Crossing the anmictic oulations together forms the F shown. Random mating the F gives rise to the F generation, the oulation in which genetic effects are described in Willham and Pollak (985). Inbreeding oulations and to comlete homozygosity generates the oulations with the genotyic arrays shown. Crossing these two inbred oulations roduces the same F as crossing the two anmictic oulations. generation is obtained is given in Fig.. Assuming two alleles er locus, the generation means are F F P ( f ) ( f ) ( f ) P ( f ) = ( = ( = ( = ( f )( F f )( F f )( F f )( F + d) + ) + fa( fa( + α d) + α d) + fa( fa( where f = inbreeding coefficient of a generation; i = frequency of the = frequency of the i i = i ), ), ith allele in oulation ; ith allele in oulation ; + ), and ), + i = average allele frequency in the cross of oulation and ; and

4 Lamkey and Edwards - 4 δ δ i i = = one half the difference in allele frequency between oulations. In the two allele case, a = half of the difference between homozygotes; α = a + d( F i = δ = a( = ; and d = the deviation of the heterozygote from the homozygote midarent; ) = average effect of an allele substition; ) + d = mean of F generation. Panmictic-midarent heterosis is the heterosis observed when two random mating oulations are crossed to form an F hybrid. The strict definition of anmictic-midarent heterosis is the difference between the mean of the F and the average of the two random mated arent oulations (midarent value). F heterosis is defined as the difference between the mean of the F generation and the midarent value. Algebraically, these heterosis values are: Panmictic-midarent heterosis = 4 d, and F heterosis = d. Four conclusions can be drawn from these exressions: ) heterosis is deendent on directional dominance; ) heterosis is a function of the square of the difference in allelic frequency between two oulations and therefore, heterosis is secific to a articular cross; 3) if two inbred lines are crossed, can only be 0 or, therefore, heterosis in a cross of two inbred lines is a function of dominance at those loci that carry different alleles in the inbred lines (Falconer and Mackay, 996); and 4) randomly mating the F reduces heterosis by 50%. Although genetic divergence (difference in allelic frequency) and dominance are necessary for there to be heterosis, they are not sufficient in the case of multile alleles. Cress (966) showed that with multile alleles segregating in a oulation the lack of heterosis cannot be used to infer a lack of genetic divergence between the arental oulations. This result has imortant imlications when breeders are screening oulations to establish new heterotic grous. Falconer and Mackay (996) refer to heterosis as the converse of inbreeding deression. This is sometimes an overlooked fact and reresents one of the breakthroughs in the discovery of heterosis. This fact is articularly relevant to the inbred-hybrid system of breeding in which heterosis is generally calculated by using the mean of inbred arents, as oosed to the mean of random mated oulations, as in Falconer and Mackay (996). We define inbred-midarent heterosis as the difference between the mean of the F and the mean of the arent oulations when inbred to homozygosity. The vigor lost during inbreeding of the arent oulations is restored in the F. This gives rise to the concet of baseline heterosis. Baseline heterosis is simly the restoration of what was lost because of inbreeding deression. Inbred-midarent heterosis is equal to baseline heterosis lus anmictic-midarent heterosis. Thus, baseline heterosis is equal to inbredmidarent heterosis minus anmictic-midarent heterosis, which is equal to the difference between the anmictic midarent value and the inbred midarent value, or simly the average inbreeding deression observed in the two anmictic arent oulations. Algebraically, Inbred-midarent heterosis = d + d, and d d. baseline heterosis = Note that inbred-midarent heterosis is a function of inbreeding deression, genetic divergence, and dominance whereas anmictic-midarent heterosis is a function

5 Lamkey and Edwards - 5 only of genetic divergence and dominance. Surrisingly little attention has been given to eistasis and heterosis. In the 950 conference on heterosis (Gowen, 95), eistasis as a cause of heterosis was not directly addressed. Willham and Pollak (985) resented heterosis theory for the case of two linked loci with eistasis. Although the equations are too comlicated to model, a coule of imortant oints emerge concerning eistasis and heterosis. Panmictic-midarent heterosis is a function of dominance and unlinked additive x additive eistasis at loci with genetic divergence. The erformance of the F hybrid, however, is a function of dominance and unlinked dominance x dominance eistasis at those loci showing genetic divergence. These observations have imortant imlications for the genetic interretation of the midarent heterosis observed in self-ollinated cros that show little inbreeding deression. The heterosis observed may be due rimarily to additive x additive eistasis, which does not contribute to inbreeding deression. GENE ACTION AND HETEROSIS Theory Single locus heterosis theory couled with the detrimental effect of Dominance a) a = 0.5, d = 0.5 b) d) c) e) Fig. 3. Plots of mean erformance and heterosis versus frequencies of the dominant allele in arent oulations for the case of comlete dominance. a) F hybrid erformance, b) midarent mean for the cross of two anmictic oulations, c) midarent mean for the cross of two oulations that have been inbred to f =, d) anmitic-midarent heterosis, and e) inbred-midarent heterosis.

6 Lamkey and Edwards - 6 Overdominance a) a = 0, d = b) d) c) e) Fig. 4. Plots of mean erformance and heterosis versus allele frequency in the arent oulations for the case of ure over dominance. a) F hybrid erformance, b) midarent mean for the cross of two anmictic oulations, c) midarent mean for the cross of two oulations that have been inbred to f =, d) anmitic-midarent heterosis, and e) inbred-midarent heterosis. recessiveness led to two rominent theories of heterosis called the dominance and overdominance hyotheses (Crow, 95). Heterosis under the dominance hyothesis is roduced by the masking of deleterious recessives in one strain by dominant or artially dominant alleles in the second strain. Heterosis under the overdominance hyothesis is due to heterozygote sueriority and, therefore, increased vigor is roortional to the amount of heterozygosity. Suorters of the overdominance hyothesis ut forth two main objections to the dominance hyothesis. First, it should be ossible to accumulate by selection all the favorable dominance alleles into one homozygous strain and obtain inbreds that are as vigorous as hybrids. Second, F distributions should be skewed because of the ¾ dominants to ¼ recessives segregation. Jones (97) showed with linkage and Collins (9) showed with large numbers of loci that the overdominance and dominance hyotheses were essentially indistinguishable. Crow (948) using a mutation-selection equilibrium argument felt that dominance was insufficient to exlain heterosis in maize. Hull (95) resented eight reasons why he felt overdominance was the cause of heterosis in maize. The debate over the tye of gene action controlling heterosis has gone on for more that 80 years. As we will see later in the section entitled gene action, this debate had a major influence on the develoment of breeding methodology.

7 Lamkey and Edwards - 7 b) Fig. 5. Plots of baseline (solid lines) and functional heterosis (dashed lines) versus allelic frequencies in the arents. Inbred-midarent heterosis is the sum of anmicitic-midarent heterosis and baseline heterosis. Fig. a) is for the case of comlete dominance and b) is for the case of ure overdominance. We have alied our extension of the theory resented by Willham and Pollak (985) to the cases of dominance and overdominance to further illustrate some key rinciles. In the case of comlete dominance (Fig. 3), several key oints are obvious. ) F erformance is maximized when the favorable allele is fixed in one of the oulations. ) With one locus the best inbred is as good as the best hybrid. 3) Panmictic-midarent heterosis is maximized as the midarent values are declining. 4) When inbred oulations are crossed, heterosis exists even in the absence of genetic divergence. Pure overdominance is similar to comlete dominance (Fig. 4), but the major excetion is that it is not ossible to obtain an inbred line as good as a hybrid with overdominance. Baseline and anmictic heterosis for dominance and overdominance are lotted in Fig. 5. The grahs are very similar for the two tyes of gene action with the major difference being the magnitude; more heterosis is observed with overdominance than with dominance. The second and most imortant oint is that anmictic-midarent heterosis only exceeds baseline heterosis when allelic frequencies are at the extremes. This is an imortant oint to kee in mind when studying heterosis among inbred lines. A significant ortion of the heterosis among inbred lines is due simly to recovery of what was lost during inbreeding and in some instances little of the observed heterosis may actually be due to genetic divergence. Emirical Studies Gene action and gene effects have been extensively studied in many cro secies. Gene action is imortant in determining cultivar tye (hybrid, ure line, synthetic, etc.), breeding methodology used to develo cultivars, and in the interretation of quantitative genetic exeriments. The study of gene action has been aroached in two ways (Srague, 966). One characterizes the redominant tyes of genetic variance (additive vs. dominant) in oulations, an activity that lead to develoment and analysis of mating designs, including the North Carolina mating designs (see Hallauer and Miranda, 988 for a review). Because of the difficulties in artificial hybridization, the variance comonent aroach is not used frequently in self-ollinated cros; instead generation mean analysis has been the most rominent aroach to determining gene action in these secies. The results of these studies lead to the roosal of many breeding methods that caitalize on different tyes of gene action, including recurrent selection for general combining

8 Lamkey and Edwards - 8 ability and inbred er se selection (additive effects), recurrent selection for secific combining ability (dominance effects), and recirocal recurrent selection (both additive and dominance effects). Of the major cro secies, gene action has been most extensively studied in maize. A review of gene action studies in maize is therefore aroriate. Four lines of evidence will be reviewed: variance comonent estimation, generation means analysis, recurrent selection, inbreeding deression, and measured genotyes studies. Srague and Eberhart (977), Gardner (963), and Hallauer and Miranda (988) have excellent reviews of gene action studies in maize. Our review will be confined to grain yield. Variance Comonent and Generation Means Studies Numerous variance comonent estimation studies have been conducted in maize. Hallauer and Miranda (988) reviewed and summarized variance comonent studies in maize conducted through the mid 980s. The general conclusion from these studies is that in most maize oulations, additive genetic variance for grain yield is usually to 4 times larger than dominance variance. Dominance variance is imortant in maize oulations and often is significant, but it is usually much smaller than additive variance. These results are often interreted as imlying that additive effects are of rimary imortance for grain yield of maize and that grain yield is controlled by genes with artial to comlete dominance. The Design III mating design was develoed secifically to estimate the degree of dominance (Comstock and Robinson, 95) by utilizing F oulations, in which the allele frequency is 0.5. Several Design III exeriments have been conducted. Estimates of degree of dominance from F oulations were usually in the overdominant range. These scientists realized from the outset that reulsion hase linkage would bias degree of dominance uward, so exeriments were develoed to reduce the linkage bias by random mating the F oulations. Estimates of average degree of dominance estimated from random mated F oulations were always smaller than the estimates from nonrandom mated F oulations and usually in the artial to comlete dominance range. These results convinced all but the most adamant overdominance suorters that much of the observed overdominance was robably due to linkage bias. The early variance comonent studies assumed that eistasis was unimortant for grain yield of maize. This assumtion was required because the number of covariances of relatives were not available to estimate eistasis and because eistatic models are difficult to handle mathematically. From both a breeding methodology and statistical oint of view, eistasis is difficult to estimate. Studies estimating eistasis in maize are too numerous for comrehensive review (see Hallauer and Miranda, 988 for an excellent review), but a few interesting conclusions can be drawn. Studies estimating eistasis by generation means analysis generally have reorted significant eistatic effects. Estimates made by the analysis of variance (covariance of relatives) aroach generally have reorted nonsignificant eistatic effects. Studies with oenollinated varieties generally have shown additive effects to be more imortant than dominance or eistatic effects, and studies with elite inbred lines generally have reorted dominance and eistatic effects to be more imortant than additive effects. These results are interesting and ambiguous at best. The lack of detection of eistasis with variance comonent studies suggests either a lack of statistical ower or that eistasis is relatively unimortant. The ability to detect eistasis with generation means studies is indicative of the greater statistical ower of using means, but these studies usually have a narrow inference base. Generally, it has been acceted that eistasis is relatively unimortant, but that there may be secific hybrid combinations in which eistasis is imortant. These conclusions are

9 Lamkey and Edwards - 9 interesting considering the findings from molecular biology during the ast 5 years. It is well known now that genes at the molecular level interact with each other or exhibit eistasis (Coe et al., 988). The question is: why have we been relatively unsuccessful at detecting eistasis at the henotyic level? The difficulty in detecting eistasis in oulations at the henotyic level, desite its ubiquitous resence at the molecular level, may be related mostly to an inadequate understanding of eistasis at the oulation level. Geneticists have long known about eistasis, but their concet of eistasis (hysiological eistasis) is different from a quantitative geneticist s statistical or oulation eistasis. Physiological eistasis occurs when henotyic differences among individuals with various genotyes at one locus deends on their genotyes at another locus (Cheverud and Routman, 995). Statistical eistasis is a deviation of multilocus genotyic values from the additive combination of their single locus comonents (Cheverud and Routman, 995). The main distinction between these two definitions is that statistical eistasis is a oulation henomenon deendent on allelic frequencies in a secific oulation, whereas hysiological eistasis is a genotyic henomenon indeendent of allelic frequencies at the loci in question (Cheverud and Routman, 995), Cheverud and Routman (995) demonstrated that additivity (a), dominance (d), and eistasis (e) all contribute to the average effects of alleles and the additive genetic variance. Only dominance and eistasis contribute to dominance deviations and variance, and eistasis alone contributes to the eistatic interaction deviations and variance. This means that hysiological eistasis makes imortant contributions to additive and dominance variance and only the remainder contributes to statistical eistasis. It is also imortant to note that this concet is different from the confounding of statistical eistasis with additive and dominance genetic variances as often haens in one and two factor mating designs. Cheverud and Routman (995) were able to show that hysiological eistasis can either suress or enhance additive and dominance genetic variance. In some instances, deending on allelic frequencies, genetic variances and in articular dominance variance, can be suressed or enhanced u to 50%. Using this same two locus aroach and varying allelic frequencies at the two loci, Cheverud and Routman (996) set u models with only additive-by-additive, additive-by-dominance, and dominance-by-dominance eistasis. Additive (a) and dominance (d) genotyic values for these models were zero. They were able to show that under certain allele frequencies that additive and dominance genetic variance exists in these oulations. In essence, they along with others have shown that with finite oulations, eistasis can contribute to the additive genetic variance. Results from generation means analyses have been more ambiguous. Hallauer and Miranda (988) reviewed the advantages and disadvantages of generation means models. Generally, two tyes of generation means studies have been conducted. One tye involves a diallel among a grou of inbred lines or oulations. For diallel studies, models of Griffing (956), Eberhart and Gardner (966), and Gardner and Eberhart (966) are often used. With these studies, the reference or inference oulation is restricted to the set of lines or oulations included in the study. Tyically, only general and secific combining ability effects can be estimated, although in the more advanced models of Eberhart and Gardner (966) eistatic effects can be estimated as well. The second tye of generation means analysis involves the cross between two inbred lines and generations derived from this cross (e.g., F, backcrosses to the arents). These studies are even more restricted in their inference base and have been used mostly for studying the inheritance of secific traits. Several methods are available for analyzing these tyes of studies (See Hallauer and Miranda, 988 for a review). Classical generation means studies involving inbred lines and derived generations tyically have the F as the inference oulation. This is a

10 Lamkey and Edwards - 0 disadvantage in cros exhibiting heterosis, because the inference oulation is not reflective of elite maize hybrids. Melchinger (987) roosed a generation means model to analyze testcrosses of generations derived from two inbred lines. The reference oulation for this model is the F testcross oulation in gametic hase equilibrium, which is more directly alicable to elite maize hybrids. He develoed models for means and variances that included linkage and eistasis. The tyical genetic design for Melchinger s model involves choosing two inbreds from the same heterotic grou (P and P) and an inbred from the oosite heterotic grou (PT). P and P are used to generate F, F, BCP, and BCP generations. In addition, to enhance the ower of the model, an F generation can be develoed by selfing the F to homozygosity or the F generation can be random mated for several generations (See Melchinger, 987 and Lamkey et. al., 995). Each of these generations is testcrossed to PT and generation means analysis is calculated by using the testcross generation means. Variances of each of the segregating generations can also be analyzed by crossing individual lants from each of the generations onto the tester, PT. Because linkage has no effect on the means in the absence of eistasis, only two models need to be fit to the data. Model allows for linkage, but not eistasis: where M (d θ d x = coefficient that is generation deendent. Suerscrit T denotes arameters that are intrinsic to the tester used in the study. Model allows for eistasis, but not linkage: where ( i T T Y T = generation testcross mean; = + if oulation, and, T Y = m + x( d = testcross mean of the F oulation in gametic equilibrium, ) = j T j j θ j d T j P carries the favorable allele at locus = half the average effect of a gene substitution at locus j in the F testcross ) = θ θ j< k j i T k jk and T i jk = additive x additive eistatic effect between loci j and k. Lamkey et al. (995) fit Melchinger's model to the P, P, F, F -Syn 8, BCP, and BCP generations derived from the inbreds B73 (P) and B84 (P). B73 and B84 are from the same heterotic grou and are related to the extent that they were both develoed from Iowa Stiff Stalk Synthetic (BSSS). Inbred Mo7 from the Lancaster Sure Cro heterotic grou was used as the tester. The results of fitting Models and to the testcross means are shown in Fig. 6. Model, which allows linkage, but not eistasis, exlained 48% of the variation among testcross means, but had a highly significant lack of fit. Model, which allows eistasis, but not linkage, exlained 69% of the variation among testcross means and detected significant eistatic effects. These results indicated that unlinked eistatic effects accounted for % of the variation among generation means. In addition, the lack T ), T T Y = m + x( d ) + x j and - otherwise, ( i T ),

11 Lamkey and Edwards Observed Yields Model : Linkage, but no eistasis, R=48% Model : Eistasis, but no linkage, R=69% Grain Yield (t/ha) F 6.5 F-Syn P BCP F BCP P Percent B73 Fig. 6. The results of fitting Models and to the six testcross generation means. Testcross erformance is lotted against the ercentage of B73 germlasm in the oulation. From Lamkey et al. (995). of fit for Model was significant as well, indicating that other eistatic effects, both linked and unlinked, are imortant in this oulation. Favorable eistatic gene combinations have been accumulated in B73 and B84. Lamkey et al. (995) found that the genetic variance among BCP rogenies was not significant. Melchinger et al. (988) also reorted that backcrossing to the higher yielding arent resulted in a nonsignificant genetic variance comonent. These results are further evidence of the imortance of eistasis and suggest that it may not be ossible to accumulate favorable alleles for grain yield into one arent in an additive fashion as redicted by the dominance theory of heterosis (Lamkey et al., 995). This result has imortant imlications for marker assisted selection and backcrossing rograms, which rely on the additive accumulation of favorable alleles into a arent. Recurrent Selection Patterns of resonse to recurrent selection are also indicative of the tye of gene action controlling a trait. Srague and Miller (950) roosed a selection exeriment to test what tye of gene action was imortant for a trait. The remise of their method was that selection for general combining ability is made on the assumtion that dominant favorable genes are imortant in heterosis and selection for secific combining ability is made on the assumtion that overdominance and eistasis are mainly resonsible for heterosis. With selection for general combining ability, the average allele frequency for genes affecting a trait will aroach.0 as a limit. With selection for secific combining ability, the allele frequency in the oulation undergoing selection would aroach -q if the average allele frequency in the homozygous tester is q. The exeriment involves choosing two oulations A and B, in which selection will be racticed and an inbred line C as the tester arent. Standard half-sib selection is conducted in A and B for a number of cycles by using C as the tester. Imroved cycles of A and B are designated as A, A, etc. If selection has been rimarily fixing dominant alleles in A and B, the crosses between A x B, A x B, etc should exhibit an increase in yield relative to A x B. Similarly, A, A, etc. should be higher in yield that the original A. If selection has been to rimarily fix recessive alleles for those loci where tester C carries dominants and dominant alleles where tester C carries recessive alleles, then the crosses A x B, A x B, etc. should exhibit a downward trend relative to the original cross A x B. Trends in A, A, etc. relative to A would deend on allele frequency in the tester C.

12 Lamkey and Edwards - Table. Selection resonse in Alh and the F of (WF9 x B7) after five cycles of selection using B4 as the tester. Data from Russell et al. (973). Tester Alh Cn (WF9 x B7)Cn Poulation er se.06 ± ± 0.44 B ± ± 0.50 BSBB 3.63 ± ± 0.50 C0 of oulation er se.9 ± ± 0.44 C0 of other oulation.63 ± ± 0.8 Interoulation cross 4.09 ± 0.8 Russell et al. (973) reorted on an exeriment to comare the imortance of dominance and overdominance for yield heterosis in maize by using the rocedure of Srague and Miller (950). They conducted five cycles of selection for secific combining ability in the oen-ollinated variety Alh and the F of WF9 x B7. Resonses for grain yield for six different testers are shown in Table. They found significant increases in grain yield in both the oulations er se and in the interoulation crosses suggesting that overdominance was not imortant for grain yield in these two oulations. A significant result of this study was that selection for secific combining ability was effective for imroving general combining ability as well as evidenced by the erformance of the oulations when crossed to BSBB (a broad based oulation) as well as the C0 of the other oulation. The imlication of this study was that using a single tester would also give imrovement with other testers as well. This was further evidence in suort of the concet of early testing (Jenkins, 935; Srague, 946) that is commonly used in maize breeding today. The finding that additive effects were of rimary imortance for grain yield of maize and that overdominance was relatively unimortant increased interest in inbred rogeny selection methods and the search for high yielding inbred lines. Comstock (964) demonstrated that in the absence of overdominance S - or S - rogeny selection was exected to be suerior to other methods of recurrent selection for oulation imrovement. Inbred rogeny selection has had variable levels of success. Lamkey (99) and others (see Lamkey (99) for references) found that results from S -rogeny selection were discouraging. Several reasons could account for the lack of resonse including lack of genetic variance, overdominance, and random genetic drift. Horner et al. (989) comared S - rogeny selection with half-sib selection by using an inbred tester in two maize oulations. They found greater rates of gain for half-sib selection and concluded that nonadditive gene action in the overdominant range was imortant for grain yield in these oulations. Long-term recirocal recurrent selection (RRS) studies also rovide evidence on the tye of gene action for heterosis. Cress (967) conducted simulation studies of RRS by using both dominant and overdominant gene action models. With comlete dominance, the mean of the interoulation cross (hybrid), and the mean of the two oulations are exected to increase, excet by chance. But, with overdominance, the change in oulation mean deends on the equilibrium gene frequency. It is ossible to get short term increases in the oulation er se means, but in the long-term they always decrease. Keeratinijakal and Lamkey (993a) evaluated resonse to cycles of RRS in BSSS and Iowa Corn Borer Synthetic # (BSCB). They reorted gains of 7% er cycle in the interoulation cross, no change in BSSS er se, and a small significant increase in BSCB. The small genetic gains in the oulations er se did not resemble the resonse atterns that Cress (967) redicted for overdominance and was attributed to random genetic drift due to small effective oulation size (Keeratinijakal and Lamkey, 993b). Inbreeding deression in

13 Lamkey and Edwards - 3 BSSS decreased over cycles of selection and showed no change in BSCB, whereas inbreeding deression in the interoulation cross doubled from C0 to C. Heterosis of the interoulation cross increased from 0.86 to.9 Mg ha -. These changes in inbreeding deression and heterosis suggest that selection has been for alleles at comlementary loci in each oulation, such that the interoulation cross is becoming more heterozygous with selection. More recent molecular data seem to suort this conclusion (Labate et al., 997; 998). An analysis of genetic divergence of dominance-associated distances also indicated that overdominance was not imortant in these oulations (Keeratinijakal and Lamkey, 993b). Inbreeding Deression Inbreeding deression is the converse of heterosis. The mean of a oulation with inbreeding coefficient f is: Mf = M0 f d q where summation is over all loci controlling a trait and and q are the allele frequencies in the whole oulation (Falconer and Mackay, 996). Several conclusions can be drawn from this equation, like the one for heterosis. First of all, a locus will not contribute to inbreeding deression if d = 0 or there is no dominance. Second, the direction of the change in mean is toward the value of the recessive allele. Third, inbreeding deression is maximized when =q=0.5, which is also where the number of heterozygotes are maximized. Fourth, in the absence of eistasis, inbreeding is a linear function of f. Fifth, if there is eistasis, but no dominance, there will not be any inbreeding deression (Crow and Kimura, 970). Sixth, if there is eistasis and dominance, then inbreeding deression will be a quadratic or higher function of f (Crow and Kimura, 970). These basic results have several imortant imlications regarding heterosis and hybrids in self-ollinated cros. Several studies with soybean [Glycine max (L.) Merr.] have reorted significant heterosis (Burton, 987). Over all the heterosis studies that Burton reviewed, 85% of the F crosses showed midarent heterosis and 6% showed high arent heterosis. In soybean, there obviously is heterosis, but the genetic cause of the heterosis remains obscure. Heterosis alone, is not good evidence for dominance; however, heterosis studies conducted in conjunction with inbreeding deression studies should give a clear icture of the tyes of gene action involved in heterosis in soybean. For examle, if there is midarent heterosis, but no inbreeding deression then there would be good evidence for the existence of additive x additive eistasis. Numerous inbreeding deression studies have been conducted in maize. The majority of the studies have reorted a linear relationshi between inbreeding deression and f, and have concluded that eistasis is unimortant for grain yield (Hallauer and Miranda, 988). It should be realized, however, that these studies all measured oulation bulks, and hence were looking at the average over the whole oulation. To our knowledge, there is no ublished data in maize on the variation in inbreeding deression. Pray and Goodnight (995) reorted that inbreeding deression can be genetically variable among lineages within a single oulation of flour beetle (Tribolium castaneum). Variation in inbreeding deression can be due to variation in the actual level of inbreeding, ast history of inbreeding (whether inbreeding is due to the exression of deleterious recessives or overdominance), genetic drift and fixation of different alleles in different lines (Pray and Goodnight, 995). Pray and Goodnight found evidence for nonlinearity in inbreeding deression suggesting that eistasis may be imortant for some traits. They concluded that the genetic variation resent for inbreeding deression suggests that inbreeding deression may be a heritable trait.

14 Lamkey and Edwards - 4 Measured Genotyes Measured genotyes refers to the situation in which a henotye is scored on all ossible genotyes of a two or three locus system in an otherwise homogeneous genetic background. These tyes of studies rovide considerable ower in estimating genetic effects. The disadvantage of these studies, however, is that only two or three loci can be studied at a time. Measured genotye studies may offer us the best oortunity for doing detailed studies of gene effects. Conducting a measured genotye study requires two features. First, you need genes controlling traits that you are interested in and second, you need a method of creating the aroriate genotyes in an isogenic background. The only technique for doing this in lants is backcrossing. Backcrossing has the usual roblem of linkage drag, but if the drag is the same for all gene combinations, then the bias may not be too severe. In Drosohilia for examle, the aroriate genotyes can be created in identical genetic backgrounds without backcrossing (Clark and Wang, 997). More recently, in lants, data from QTL maing studies have been used like a measured genotye study rimarily to estimate eistasis. For the sake of simlicity, only two measured genotye studies will be discussed. One from maize and one from Drosohilia. Together, these two studies bring out the salient features of the analyses. Russell (976) develoed B4 isolines of the 7 genotyes ossible for three loci with two alleles. The exeriment was grown for three years at one location and data were collected for 0 traits. The standard Cockerham model was fit to the data to estimate additive, dominance, and eistatic effects. Russell (976) found that 87, 7, 47, 5, 3, and 30% of the additive, dominance, additive x additive, additive x dominance, and dominance x dominance effects were significant at the 5% level, indicating that even at the oulation level, loci were interacting fairly frequently. This study also demonstrates the leiotroic effect of loci. Clark and Wang (997) reorted on a measured genotyes study in Drosohila. They constructed all ossible two locus genotyes for each of 8 airs of P-element insertions. They found significant eistatic effects in 7% of their comarisons. They alied the method of Cheverud and Routman (995) and found significant hysiological eistasis in 5% of the comarisons. Clark and Wang reorted eistatic effects on the same order of magnitude as main effects. These studies clearly demonstrate that genes interact. Measured genotyes using a combination of Cockerham s analysis and Cheverud and Routman s analysis, may be one of the best tools for understanding eistasis and its contribution to heterosis. NEEDS FOR FUTURE RESEARCH The data clearly indicate a need for future emirical and theoretical research into heterosis; however, we need to be very careful about how future exeriments are designed and analyzed. Enfield (977) was critical of emirical quantitative genetic exeriments indicating that the literature was either cluttered with exeriments that were meaningless because of standard errors that were too large or because standard errors were absent altogether. Desite the tremendous develoments that have recently occurred in molecular biology, quantitative genetics is still the only theory linking genotye to henotye. It is imerative that we design better exeriments to test the adequacy and validity of quantitative genetic models. We would like to roose several areas of research that are needed to better understand heterosis. We will not resent exerimental aroaches, because we do not have all of the answers. ) Gene action and effects are key to understanding the inheritance of

15 Lamkey and Edwards - 5 quantitative traits. For maize at least, it seems that from average oulation estimates, there is no evidence for overdominance. Although this is useful information, what would be even better is to gain insight into the distribution of gene effects and gene action for individual traits. Are there lots of loci with equal and small effects or is there some tye of distribution? The conventional aroach to gene action studies will not answer this question and new aroaches will be needed. ) Selection exeriments in lants need to be better designed. Most of our current recurrent selection exeriments are not adequately designed to searate the effects of selection from drift, so it becomes nearly imossible to reliably interret the results of these exeriments. It is clear that recurrent selection works, and future exeriments to demonstrate the effectiveness of recurrent selection are robably not needed. But we do need well-designed exeriments with adequate controls and relication. New selection exeriments should either be relicated or include an unselected, relicated control oulation of the same effective size. 3) The view of random genetic drift in agricultural is the one that drift leads to a loss of heterozygosity and eventual erosion of genetic variance. Although this is true in the additive gene action case, with dominance and eistasis, drift may reduce heterozygosity with a corresonding increase in additive genetic variance. We need to incororate this new information from evolutionary biology into the design of our breeding rograms. 4) Eistasis has long been ignored in breeding rograms and is generally assumed to be absent or unimortant. Evidence from molecular biology clearly shows that genes interact. Recent theory from evolutionary biology that distinguishes hysiological eistasis from oulation eistasis may indicate why oulation level eistasis may be undetectable. More theoretical work is needed to otimally design breeding rograms to select for eistatic effects. 5) Desite several classic studies, we know very little about inbreeding deression in lants. Because much of the observed heterosis among inbred lines may be due to the recovery of inbreeding deression, the genetics of heterosis may be best elucidated by studying the genetics of inbreeding deression. Our reoccuation with heterosis has caused us to overlook the imortance of inbreeding deression. There are of course several roblems in designing and conducting good quantitative genetics exeriments. First, they are often large and consume considerable hysical resources, even for lab secies. Second, there seems to be little funding available in agricultural secies to do quantitative genetics and lant breeding related research. Third, there are few scientists being trained to do this tye of research. Lack of funding and suort in quantitative genetics may in the long term severely limit future genetic gains. ACKNOWLEDGEMENTS Joint contribution from the Corn Insects and Cro Genetics Research Unit, Agricultural Research Service, USDA, Det. of Agronomy, Iowa State Univ. and Journal Paer No. J-777_of the Iowa Agric. and Home Economics Ex. Stn. Project No. 3495, and suorted by Hatch Act and State of Iowa.

16 Lamkey and Edwards - 6 REFERENCES Burton, J. W Quantitative genetics: results relevant to soybean breeding In: Soybeans: imrovement, roduction and uses. J. R. Wilcox (ed.), ASA, CSSA, SSSA, Madison, WI. Cheverud, J. M. and E. J. Routman Eistasis and its contribution to genetic variance comonents. Genetics 39: Cheverud, J. M. and E. J. Routman Eistasis as a source of increased additive genetic variance at oulation bottlenecks. Evolution 50: Clark, A. G. and L. Wang Eistasis in measured genotyes: Drosohila P- element insertions. Genetics 47: Coe, E. H. Jr., M. G. Neuffer, and D. A. Hoisington The genetics of corn In: G. F. Srague and J. W. Dudley (eds.) Corn and Corn Imrovement. Am. Soc. Agron., Madison, WI. Comstock, R.E., 964. Selection rocedures in corn imrovement In: W. Heckendorn and J. I. Sutherland (ed.) Reort 9th Hybrid Corn Industry-Res. Conf., 9-0 Dec American Seed Trade Association, Washington, D. C. Comstock, R. E. and H. F. Robinson. 95. Estimation of the average dominance of genes In: Heterosis J. W. Gowen (ed.). Iowa State College Press, Ames. Collins, G. N. 9. Dominance and vigor of first generation hybrids. Am. Nat. 55:6-33. Cress, C. E Heterosis of the hybrid related to gene frequency differences between two oulations. Genetics 53: Cress, C. E Recirocal recurrent selection and modifications in simulated oulations. Cro Sci. 7: Crow, J. F Alternative hyothesis of hybrid vigor. Genetics 33: Crow, J. F. 95. Dominance and overdominance In: Heterosis J. W. Gowen (ed.). Iowa State College Press, Ames. Crow, J. F. and M. Kimura An introduction to oulation genetics theory. Burgess, Minneaolis, Minnesota. Eberhart, S. A. and C. O. Gardner A general model for genetic effects. Biometrics : Enfield, F. D Selection exeriments in Tribolium designed to look at gene action issues In: E. Pollak, O. Kemthorne, and T. B. Bailey, Jr. (eds.) Proceedings of the International Conference on Quantitative Genetics, Ames, IA. 6- Aug Iowa State University Press, Ames. Falconer, D. S. and T. F. C. Mackay Introduction to quantitative genetics. 4th ed. Longman, Essex, England

17 Lamkey and Edwards - 7 Gardner, C. O Estimates of genetic arameters in cross-fertilizing lants and their imlications in lant breeding In: Statistical genetics and lant breeding. W. D. Hanson and H. F. Robinson (eds.), NAS-NRC Publ. 98. Gardner, C. O. and S. A. Eberhart Analysis and interretation of the variety cross diallel and related oulations. Biometrics : Griffing, B Concet of general and secific combining ability in relation to diallel crossing systems. Australian J. Biol. Sci. 9: Gowen, J. W. (ed.) 95. Heterosis. Iowa State College Press, Ames. Hallauer, A. R. and J. B. Miranda, Fo Quantitative genetics in maize breeding. nd ed., Iowa State Univ. Press, Ames. Horner, E.S., E. Magloire, J.A. Morera Comarison of selection for S rogeny vs. testcross erformance for oulation imrovement in maize. Cro Sci. 9: Hull, H. F. 95. Recurrent selection and overdominance In: Heterosis J. W. Gowen (ed.). Iowa State College Press, Ames. Jenkins, M. T The effect of inbreeding and of selection within inbred lines of maize uon the hybrids made after successive generations of selfing. Iowa State Coll. J. Sci. 9: Jones, D. F. 97. Dominance of linked factors as a means of accounting for heterosis. Genetics : Keeratinijakal, V. and K. R. Lamkey. 993a. Resonses to recirocal recurrent selection in BSSS and BSCB maize oulations. Cro Sci. 33: Keeratinijakal, V. and K. R. Lamkey. 993b. Genetic effects associated with recirocal recurrent selection in BSSS and BSCB maize oulations. Cro Sci. 33:78-8. Labate, J. A., K. R. Lamkey, M. Lee, and W. L. Woodman Genetic diversity after recirocal recurrent selection in BSSS and BSCB maize oulations. Cro Sci. 37: Labate, J. A., K. R. Lamkey, M. Lee, and W. L. Woodman Poulation genetics of increased hybrid erformance between two maize oulations under recirocal recurrent selection. (This volume) Lamkey, K. R. 99. Fifty years of recurrent selection in the Iowa stiff stalk synthetic maize oulation. Maydica 37:9-8. Lamkey, K. R., B. S. Schnicker, and A. E. Melchinger Eistasis in an elite maize hybrid and choice of generation for inbred line develoment. Cro Sci. 35:7-8. Melchinger, A.E Exectation of means and variances of testcrosses roduced from F and backcross individuals and their selfed rogenies. Heredity 59:05-5.

18 Lamkey and Edwards - 8 Melchinger, A.E., W. Schmidt, and H.H. Geiger Comarison of testcrosses roduced from F and first backcross oulations in maize. Cro Sci. 8: Pray, L. A. and C. J. Goodnight Genetic variation in inbreeding deression in the red flour beetle Tribolium castaneum. Evolution 49: Russell, W. A Genetic effects and genetic effect x year interactions at three gene loci in sublines of a maize inbred line. Can. J. Genet. Cytol. 8:3-33. Russell, W. A., S. A. Eberhart, and O. U. A. Vega Recurrent selection for secific combining ability for yield in two maize oulations. Cro Sci. 3:57-6. Shull, G. H. 95. Beginnings of the heterosis concet In: Heterosis J. W. Gowen (ed.). Iowa State College Press, Ames. Srague, G. F Early testing of inbred lines of corn. J. Am. Soc. Agron. 38:08-7. Srague, G. F Quantitative genetics in lant imrovement In: Plant Breeding, K. J. Frey (ed.) Iowa State University Press, Ames. Srague, G. F. and S. A. Eberhart Corn Breeding In: G. F. Srague (ed.) Corn and Corn Imrovement. nd ed. Agron. Monogr. 8. ASA, Madison, WI. Srague, G. F. and P. A. Miller A suggestion for evaluating current concets of the genetic mechanism of heterosis of corn. Agron. J. 4:6-6. Willham, R. L. and E. Pollak Theory of heterosis. J. Dairy Sci. 68:4-47.