Heritability estimates of citral content in East Indian lemongrass

Summary Citral content in oil obtained from 80 representative clones, chosen randomly from a population raised from open-pollinated seeds of clonally propagated East Indian lemongrass variety SD-68 and their half-sib progenies was estimated by gas chromatography. Heritability estimates were determined by variance component and parent-offspring regression analyses. Broad-sense heritability was 0.49 while narrow sense heritabilities by variance and regression analyses were 0.37 and 0.24, respectively. Phenotypic and genotypic correlations between citral content in oil and oil content were very low both among the parental clones (-0.01 and -0.01, respectively) and among their progenies (0.13 and 0.08, respectively) indicating that selection for either of these traits would not have much effect on the other.     

Path analysis in fruits of the buriti (<Emphasis Type="Italic">Mauritia flexuosa</Emphasis> L. f) in populations from the Brazilian Middle North Region

The Mauritia flexuosa L.f. tree is of immense socioeconomic significance in the Brazilian Middle North Region for its manifold benefits, but mainly for its fruit. However, the potential of this species has still not been extensively studied. The objectives of this work were to study the phenotypic and genotypic correlations between the characteristics of the M. flexuosa fruits and to establish the direct and indirect effects of the secondary characteristics of the fruit on the pulp yield. Samples of ten fruits per genotype were gathered from four natural populations, from 240 different genotypes. These samples were assessed in terms of the fruit, almond and hull weights, equatorial and polar fruit diameters, and the fruit, almond and pulp, volumes, as well as the pulp yield. The genotypic correlations showed greater magnitude, higher than the phenotypic correlations, and both were significant and revealed equal signs. The indirect selection of the M. flexuosa genotypes with greater fruit weight and almond volume favored the increase in the pulp yield in this fruit species.     

Social cues affect quantitative genetic variation and covariation in animal personality traits

The social environment is expected to have substantial effects on behavior, and as a consequence, its heritability and evolvability. We investigated these effects by exposing Australian field crickets (Teleogryllus oceanicus) to either silence or recordings of male acoustic sexual signals. We used a combined pedigree and full‐sib/half‐sib breeding design to estimate the repeatability, heritability, and evolvability of behaviors related to boldness, exploration, and activity. All behaviors measured were significantly repeatable in both social environments. Additionally, most behaviors showed significant heritabilities in the two environments. We found no difference in repeatabilities between the silent and the acoustic environment but did find significant differences in the heritabilities and evolvabilities between these environments. There was a high degree of similarity between the phenotypic covariance matrices across the two environments, while the genotypic covariance matrices were highly dissimilar. Reflecting this, we found significant genotype‐by‐environment interactions for most of the behaviors. Lastly, we found that the repeatable aspect of behavior (“personality”) was significantly heritable for most behaviors, but that these heritabilities were higher in the acoustic than in the silent environment. We conclude that the social environment can have a significant impact on the heritability and evolvability of behavior, and argue that evolutionary inferences from phenotypic studies should be made with caution.     

Estimating heritability using family-pooled phenotypic and genotypic data: a simulation study applied to aquaculture

Estimating heritability based on individual phenotypic and genotypic measurements can be expensive and labour-intensive in commercial aquaculture breeding. Here, the feasibility of estimating heritability using within-family means of phenotypes and allelic frequencies was investigated. Different numbers of full-sib families and family sizes across ten generations with phenotypic and genotypic information on 10 K SNPs were analysed in ten replicates. Three scenarios, representing differing numbers of pools per family (one, two and five) were considered. The results showed that using one pool per family did not reliably estimate the heritability of family means. Using simulation parameters appropriate for aquaculture, at least 200 families of 60 progeny per family divided equally in two pools per family was required to estimate the heritability of family means effectively. Although application of five pools generated more within- and between- family relationships, it reduced the number of individuals per pool and increased within-family residual variation, hence, decreased the heritability of family means. Moreover, increasing the size of pools resulted in increasing the heritability of family means towards one. In addition, heritability of family mean estimates were higher than family heritabilities obtained from Falconer’s formula due to lower intraclass correlation estimate compared to the coefficient of relationship.Subject terms: Genome evolution     

The quantitative genetics of two life history trade-offs in the yellow dung fly in abundant and limited food environments

The trade-offs between body size and development time and between egg size and egg number (clutch size) are central to life history theory, but evidence for them, particularly in terms of genetic correlations, is equivocal. For the yellow dung fly Scathophaga stercoraria (Diptera: Scathophagidae), we investigated variation in phenotypic and genetic variances and covariances, i.e. heritabilities and genetic correlations, of these life history traits (plus diapause) in benign and stressful larval field or adult laboratory food environments. We found both trade-offs to be weak, as evidenced by low phenotypic and genetic correlations, but stronger in the food limited environments. Broad sense heritabilities were generally significant for all traits considered, whereas the narrow sense heritabilities for egg and clutch size were nil. With regard to the question of how environmental stress affects heritabilities, we found a whole range of responses within one single species depending on the traits considered. All three possible patterns occurred, i.e. increased h² due to increased V_G or decreased decreased h² due to increased and no change in h² due to increased V_G and V_P. These can be explained by the particular ecological circumstances yellow dung flies face in their natural environment. Nevertheless, the majority of patterns was consistent with the idea that stressful conditions amplify phenotypic differences between genotypes. Such variable responses of traits even within one organism underscores the complexity of this issue and may well explain the multiple patterns found in various organisms.Co-ordinating editor: Leimar     

Variance,Inter-Trait Correlation,Heritability, and Marker-Trait Association of Rubber Yield-Related Characteristics in <Emphasis Type="Italic">Taraxacum kok-saghyz</Emphasis>

Rubber dandelion (Taraxacum kok-saghyz or TK) is a potential industrial crop species that can produce high-quality natural rubber in its roots. The present study estimated trait variance, inter-trait correlation, and entry-mean heritability for rubber yield-related traits and analyzed associations between these traits and 42 single-nucleotide polymorphism (SNP) markers. A trial was conducted at three environments to assess a biparental progeny of 66 F₁ full-sibs, in a randomized complete block design (RCBD) with two replicates. Significant correlations, broad ranges of variation, and significant genotypic variance components were identified for five measured phenotypic traits. Moderate broad-sense heritability on an entry-mean heritability estimates (0.51–0.61) were obtained for five rubber yield-related traits based on a 1-year trial. However, the broad-sense heritability in general sense ranged from 0.09 to 0.15 depending on the trait. Two linkage groups were identified. Association analysis identified seven significant marker-trait gene associations, and only one marker was related to two traits. The implications of trait correlations and heritability for selection and improvement are discussed.     

The beta-binomial model for estimating heritabilities of binary traits

Calculations of individual narrow-sense heritability and family mean heritability of a binary trait in stochastically simulated sib trials in completely randomized block experiments showed that in some situations estimates of realized heritabilities obtained from the mixed linear threshold model could be improved by application of a proposed beta-binomial model. The proposed model adopts the beta-binomial as the conjugate-prior for the distribution of probabilities of observing the binary trait in a genetic entry. Estimation of the beta parameters allows an estimation of selection response and, by linkage to a threshold model for the individual observations, the desired heritabilities can be obtained. The average bias in the betabinomial estimates of heritability and family mean heritability was less than 2%. Improvements over existing procedures were especially manifest at heritabilities above 0.3 and at low overall probabilities of observing the trait (p < 0.30). The lowest root mean square errors were consistently obtained with the algorithm proposed by Harville and Mee (1984). The beta-binomial framework, although restricted to a single random additive genetic effect, further facilitates general analysis, estimation of selection response, and calculation of reliable family mean heritability. Intraclass correlations can be estimated directly from the beta-binomial parameters.     

Molecular marker-assisted selection for malting quality traits in barley

Selection for malting quality in breeding programs by micromalting and micromashing is time-consuming, and resource-intensive. More efficient and feasible approaches for identifying genotypes with good malting quality would be highly desirable. With the advent of molecular markers, it is possible to map and tag the loci affecting malting quality. The objective of this study was to assess the effectiveness of molecular marker assisted selection for malting quality traits. Two major quantitative trait loci (QTL) regions in six-row barley for malt extract percentage, -amylase activity, diastatic power, and malt -glucan content on chromosomes 1 (QTL1) and 4 (QTL2) have been previously identified. The flanking markers, Brz and Amy2, and WG622 and BCD402B, for these two major QTL regions were used in marker-assisted selection. Four alternative selection strategies; phenotypic selection, genotypic selection, tandem genotypic and phenotypic selection, and combined phenotypic and genotypic selection, were compared for both single and multiple trait selection in a population consisting of 92 doubled haploid lines derived from Steptoe × Morex crosses. Marker assisted selection for QTL1 (tandem genotypic and phenotypic selection, and combined phenotypic and genotypic selection) was more effective than phenotypic selection, but for QTL2 was not as effective as phenotypic selection due to a lack of QTL2 effects in the selection population. The effectiveness of tandem genotypic and phenotypic selection makes marker assisted selection practical for traits which are extremely difficult or expensive to measure such as most malting quality traits. It can substantially eliminate undesirable genotypes by early genotyping and keeping only desirable genotypes for later phenotypic selection.     

A multi-marker test based on family data in genome-wide association study

Background

Requirements for successful implementation of multivariate animal threshold models including phenotypic and genotypic information are not known yet. Here simulated horse data were used to investigate the properties of multivariate estimators of genetic parameters for categorical, continuous and molecular genetic data in the context of important radiological health traits using mixed linear-threshold animal models via Gibbs sampling. The simulated pedigree comprised 7 generations and 40000 animals per generation. Additive genetic values, residuals and fixed effects for one continuous trait and liabilities of four binary traits were simulated, resembling situations encountered in the Warmblood horse. Quantitative trait locus (QTL) effects and genetic marker information were simulated for one of the liabilities. Different scenarios with respect to recombination rate between genetic markers and QTL and polymorphism information content of genetic markers were studied. For each scenario ten replicates were sampled from the simulated population, and within each replicate six different datasets differing in number and distribution of animals with trait records and availability of genetic marker information were generated. (Co)Variance components were estimated using a Bayesian mixed linear-threshold animal model via Gibbs sampling. Residual variances were fixed to zero and a proper prior was used for the genetic covariance matrix.

Results

Effective sample sizes (ESS) and biases of genetic parameters differed significantly between datasets. Bias of heritability estimates was -6% to +6% for the continuous trait, -6% to +10% for the binary traits of moderate heritability, and -21% to +25% for the binary traits of low heritability. Additive genetic correlations were mostly underestimated between the continuous trait and binary traits of low heritability, under- or overestimated between the continuous trait and binary traits of moderate heritability, and overestimated between two binary traits. Use of trait information on two subsequent generations of animals increased ESS and reduced bias of parameter estimates more than mere increase of the number of informative animals from one generation. Consideration of genotype information as a fixed effect in the model resulted in overestimation of polygenic heritability of the QTL trait, but increased accuracy of estimated additive genetic correlations of the QTL trait.

Conclusion

Combined use of phenotype and genotype information on parents and offspring will help to identify agonistic and antagonistic genetic correlations between traits of interests, facilitating design of effective multiple trait selection schemes.     

Estimation of heritability from varietal trials data

We present the estimation of heritabilities of an observed trait in situations where evaluation of several pure breeding lines is performed in a trial at a single location and in trials from several locations. For the single location situation, we evaluate exact confidence intervals, the probability of invalid estimates, and the percentage points of the distribution of heritability. Simulations were performed to numerically verify the results. Additionally, approximations to the bias and standard error of the estimate were obtained and are presented along with their simulated values and coefficients of skewness and kurtosis. For trials in several locations, explicit expressions for exact values of confidence limits are not available. Further, one would require knowledge of one more parameter, represented by the ratio of genotype x environment (G x E) interaction variance to error variance, in addition to the number of genotypes, replication and true heritability value. Approximations were made for bias and the standard error of estimates of heritability. The evaluation of the distribution of heritability and its moments was recognized as a problem of the linear function of an independent chi-square. The methods have been illustrated by data from experiments on grain and straw yield of 64 barley genotypes evaluated at three locations.     

Laboratory Estimates of Heritabilities and Genetic Correlations in Nature

A lower bound on heritability in a natural environment can be determined from the regression of offspring raised in the laboratory on parents raised in nature. An estimate of additive genetic variance in the laboratory is also required. The estimated lower bounds on heritabilities can sometimes be used to demonstrate a significant genetic correlation between two traits in nature, if their genetic and phenotypic correlations in nature have the same sign, and if sample sizes are large, and heritabilities and phenotypic and genetic correlations are high.     

A mixed-model quantitative trait loci (QTL) analysis for multiple-environment trial data using environmental covariables for QTL-by-environment interactions, with an example in maize

Complex quantitative traits of plants as measured on collections of genotypes across multiple environments are the outcome of processes that depend in intricate ways on genotype and environment simultaneously. For a better understanding of the genetic architecture of such traits as observed across environments, genotype-by-environment interaction should be modeled with statistical models that use explicit information on genotypes and environments. The modeling approach we propose explains genotype-by-environment interaction by differential quantitative trait locus (QTL) expression in relation to environmental variables. We analyzed grain yield and grain moisture for an experimental data set composed of 976 F(5) maize testcross progenies evaluated across 12 environments in the U.S. corn belt during 1994 and 1995. The strategy we used was based on mixed models and started with a phenotypic analysis of multi-environment data, modeling genotype-by-environment interactions and associated genetic correlations between environments, while taking into account intraenvironmental error structures. The phenotypic mixed models were then extended to QTL models via the incorporation of marker information as genotypic covariables. A majority of the detected QTL showed significant QTL-by-environment interactions (QEI). The QEI were further analyzed by including environmental covariates into the mixed model. Most QEI could be understood as differential QTL expression conditional on longitude or year, both consequences of temperature differences during critical stages of the growth.     

Evolution of the male genitalia: morphological variation of the aedeagi in a natural population of <Emphasis Type="Italic">Drosophila mediopunctata</Emphasis>

To investigate the size and shape of the aedeagus of Drosophila mediopunctata, we used basic statistics and geometric morphometrics. We estimated the level of phenotypic variation, natural and laboratory heritability as well as the phenotypic correlations between aedeagus and wing measures. The wing was used as an indicator for both body size and shape. Positive significant correlation was obtained for centroid size of aedeagus and wing for field parents and their offspring reared in the laboratory. Many positive significant phenotypic correlations were found among linear measures of both organs. The phenotypic correlations were few for aedeagus and wing shape. Coefficients of variation of the measures were on average larger in the aedeagus than in the wing for offspring reared in laboratory, but not for flies coming from the field. Significant “natural” heritabilities were found for five linear measures of the aedeagus and only one for the wing. Few significant heritabilities were found for aedeagus and wing shape, mostly ones concerning the uniform components. In an exploratory analysis, we found that inversion DS-PC0 is associated with both uniform and nonuniform components of shape, respectively, in the wing and aedeagus. Our results do not support the lock-and-key hypothesis for the male genitalia evolution, but cannot refute the sexual selection and pleiotropy hypotheses.     

Characterization of variability and relationships among components of partial resistance to leaf rust in CIMMYT bread wheats

Summary A study of spring bread wheat (Triticum aestivum) germ plasm developed at the International Maize and Wheat Improvement Center (CIMMYT) showed highly significant phenotypic variability for each component of partial resistance (namely, uredial appearance period, latency period, uredial number and uredial size) to Puccinia recondita f. sp. tritici. All of the wheat genotypes displayed longer uredial appearance and latency periods and decreased uredial number and uredial size when compared to the susceptible check cultivar Morocco. Positive correlations between uredial appearance period and latency period, and uredial number and uredial size, and negative correlations between uredial appearance and latency periods and uredial number and uredial size, inclusive, suggested that the components of partial resistance were either tightly linked or under pleiotropic genetic control. Compared to Morocco, all entries had slow disease progress in the field and variation occurred in the germ plasm for the area under the leaf rust progress curve. Disease progress was negatively correlated with uredial appearance and latency periods, whereas a positive correlation was observed with uredial number and uredial size. Certain genotypes displayed high levels of partial resistance resulting in low disease incidence in the field.     

Determining the minimum sample size required to obtain sufficient progeny with a desired genotype at two quantitative trait loci

In this paper we determine the minimum progeny sample size n needed to obtain, with probability , at least m individuals of a desired two-locus genotype affecting quantitative traits. The two quantitative trait loci (QTLs) of interest may be linked or independent, with or without epistatic interaction between them. Parental genotypes may be known or unknown, and gene action at either locus may range from additive to overdominance. To reduce the required sample size, mating patterns that will produce a high proportion of desired progeny are suggested for different progeny genotypes and dominance levels. Based on the assumption of normally distributed quantitative trait expression, individuals can be classified into a genotype or genotypic group according to their phenotypic expressions. This technique is used to select both parents and progeny with unknown genotypes. Choice of parental classification criteria for a given quantitative trait affects classification accuracy, and hence the probability of obtaining progeny of the desired genotype. The complexity of this probability depends on the dominance level at each locus, the recombination fraction, and the awareness of parental genotypes. The procedure can be expanded to deal with more than two loci.BU-1168-MB in the Biometrics Unit Technical Report Series, 337 Warren Hall, Cornell University, Ithaca, NY 14853, USAFormerly known as S.-F. Shyu     

In situ wood quality assessment in Douglas-fir

The genetic control and phenotypic and genotypic correlations among wood density, modulus of elasticity, height, diameter, and volume were assessed using 967 trees representing 20 unrelated 32-year-old coastal Douglas-fir full-sib families growing on four (spaced and pruned vs. control) comparable test sites. Generally, no significant differences were observed between treatments, indicating their limited effect at assessment time. Family effect did not differ for the growth traits; however, significant differences were observed for wood density and both in situ methods (drilling resistance and acoustic velocity). Growth and wood quality attributes, individually, produced high and positive phenotypic and genetic correlations; however, high and negative correlations were observed between individual variables belonging to the two suites of attributes. Individual tree heritabilities were low for growth (0.04 to 0.08) and modest to high for wood quality attributes (0.14 to 0.68). The observed heritabilities and phenotypic and genotypic correlations imply modest to strong genetic control; however, they operated in opposing direction. The significant and consistent genetic correlations between the in situ methods and wood density and stiffness support their use as a non-destructive and economic assessment approach. The reliability of the in situ assessments was verified through cumulative pith-to-bark wood density assessment, resulting in inconsistent genetic and phenotypic correlations for early growth years. These latter findings imply that caution should be used in employing these in situ techniques as early screening tools in breeding programs.     

EVOLUTIONARY PREDICTABILITY IN NATURAL POPULATIONS: DO MATING SYSTEM AND NONADDITIVE GENETIC VARIANCE INTERACT TO AFFECT HERITABILITIES IN PLANTAGO LANCEOLATA?

Quantitative genetics has been an immensely powerful tool in manipulating the phenotypes of domesticated plants and animals. Much of the predictive power of quantitative genetics depends on the breeder's control over the context in which phenotype and mating are being expressed. In the natural world, these contexts are often difficult to describe, let alone control. We are left, therefore, with a poor understanding of the limits of quantitative genetics in natural populations. One of the crucial contextual elements for assessing breeding value is the genetic background in which an individual's genes are being assessed. When interacting genes are polymorphic within a population, the degree of mating among relatives can influence the correlations among mates and the predictions of a response to selection. Population structure can strongly influence the degree to which dominance and epistasis influences additive genetic variance and heritability. The extent of inbreeding can also influence heritabilities through its effect on the environmental component of phenotypic variance. The applicability of standard quantitative genetic breeding designs to the measurement of heritabilities in natural populations therefore depends in part on: (1) the mating system of the population; and (2) the importance of gene interactions in determining phenotypic variation. We tested for an effect of mating structure on the partitioning of phenotypic variance and heritability by comparing two breeding designs in a common environment. Both breeding designs used 139 pollen parents taken from mapped locations in a population of Plantago lanceolata L., and crossed to 280 seed parents from the same population. One design was random-mating, the second was biased toward near-neighbor matings to an extent determined by field measure of pollen-mediated gene flow distances. The offspring were grown randomly mixed in a common garden. Nine traits were measured: central corm diameter, number of leaves, area of the most recently fully expanded leaf, density of hairs (cm^-2) on the leaves, dry weight per unit leaf area, photosynthetic capacity, transpiration rates, water use efficiency, and reproductive dry weight. Heritabilities and variance components from the two designs were compared using randomization tests. None of the variance components or the heritabilities differed significantly between breeding designs at the 0.05 level. The test could distinguish differences between the heritabilities measured in the two breeding designs as small as 0.11, on average. Thus, for the degree of inbreeding normally exhibited in P. lanceolata there is insufficient gene interaction present within populations to influence the partitioning of variance between additive and nonadditive components or to influence heritability estimates to a meaningful extent. We suggest that for Plantago other sources of variation in heritability estimates, such as maternal effects and genotype × environment interactions, are more important influences than the interaction between inbreeding and gene interactions, and standard heritability estimate based on random breeding is as accurate as one taking the natural mating structure into account.     

Quantitative genetic parameters of agronomic and quality traits in a global germplasm collection reveal excellent breeding perspectives for Jatropha curcas L.

Jatropha curcas L. (jatropha) is an undomesticated plant, which has received great attention in recent years for its potential in biofuel production and in greening and rehabilitation of wastelands. Yet the absence of improved cultivars and the lack of agronomic knowledge are limiting factors for successful jatropha cultivation. The objectives of the present study were to investigate the perspectives of a worldwide jatropha breeding program and specifically to (i) estimate variance components and heritabilities for agronomic and quality traits in the early phase of cultivation; (ii) assess phenotypic and genetic correlations among those traits; and (iii) discuss strategies for breeding high yielding jatropha cultivars. Data on various traits was collected from 375 jatropha genotypes, which were tested at seven locations during the first 3 years of growth. The accumulated seed yields and the seed yields per harvest year differed significantly among the testing locations. The estimates of genetic and genotype‐by‐environment interaction variances were significant and estimates of heritabilities were high for all yield parameters. The estimates of genetic correlations indicated a strong association among yield parameters. Oil yield was strongly correlated with seed yield and only weakly with oil content in seeds. The perspectives of a jatropha breeding program are excellent. Improved cultivars, definition of favorable environmental factors and refinement of agronomic management practices are needed to secure sustainable jatropha cultivation.     

The Use of the Expectation–Maximization (EM) Algorithm for Maximum Likelihood Estimation of Gametic Frequencies of Multilocus Polymorphic Codominant Systems Based on Sampled Population Data

Estimation of gametic frequencies in multilocus polymorphic systems based on the numerical distribution of multilocus genotypes in a population sample (analysis without pedigrees) is difficult because some gametes are not recognized in the data obtained. Even in the case of codominant systems, where all alleles can be recognized by genotypes, so that direct estimation of the frequencies of genes (alleles) is possible (complete data), estimation of the frequencies of multilocus gametes based on the data on multilocus genotypes is sometimes impossible, whether population data or even family data are used for studying genotypic segregation or analysis of linkage (incomplete data). Such incomplete data are analyzed based on the corresponding genetic models using the expectation–maximization (EM) algorithm. In this study, the EM algorithm based on the random-marriage model for a nonsubdivided population was used to estimate gametic frequencies. The EM algorithm used in the study does not set any limitations on the number of loci and the number of alleles of each locus. Locus and alleles are identified by numeration making possible to arrange loops. In each combination of alleles for a given combination of m out of L loci (L is the total number of loci studied), all alleles are assigned value 1, and the remaining alleles are assigned value 0. The sum of zeros and unities for each gamete is its gametic value (h), and the sum of the gametic values of the gametes that form a given genotype is the genotypic value (g) of this genotype. Then, gametes with the sameh are united into a single class, which reduces the number of the estimated parameters. In a general case of m loci, this procedure yields m + 1 classes of gametes and 2m + 1 classes of genotypes with genotypic valuesg = 0, 1, 2,... 2m. The unknown frequencies of them + 1 classes of gametes can be represented as functions of the gametic frequencies whose maximum likelihood estimations (MLEs) have been obtained in all previous EM procedures and the only unknown frequency (P _m(m)) that is to be estimated in the given EM procedure. At the expectation step, the expected frequencies (F _m(g) of the genotypes with genotypic valuesg are expressed in terms of the products of the frequencies of m + 1 classes of gametes. The data on genotypes are the numbers (n _g) of individuals with genotypic values g = 0, 1, 2, 3, ..., 2m. The maximization step is the maximization of the logarithm of the likelihood function (LLF) for n _g values. Thus, the EM algorithm is reduced, in each case, to solution of only one equation with one unknown parameter with the use of the n _g values, i.e., the numbers of individuals after the corresponding regrouping of the data on the individuals" genotypes. Treatment of the data obtained by Kurbatova on the MNSs and Rhesus systems with alleles C, C ^w , c, D, d, E, e with the use of Weir's EM algorithm and the EM algorithm suggested in this study yielded similar results. However, the MLEs of the parameters obtained with the use of either algorithm often converged to a wrong solution: the sum of the frequencies of all gametes (4 and 12 gametes for MNSs and Rhesus, respectively) was not equal to 1.0 even if the global maximum of LLF was reached for each of them (as it was for MNSs with the use of Weir's EM algorithm), with each parameter falling within admissible limits (e.g., 0, min(P _N, P _s) for P _Ns). The ² function is suggested to be used as a goodness-of-fit function for the distribution of genotypes in a sample in order to select acceptable solutions. However, the minimum of this function only guarantee the acceptability of solutions if all limitations on the parameters are met: the sum of estimations of gametic frequencies is 1.0, each frequency falls within the admissible limits, and the gametic algebra is complied with (none of the frequencies is negative).     

Genetic analysis of certain in vitro and in vivo parameters in pigeonpea (Cajanus cajan L.)

Summary Studies on callus growth and shoots/cotyledon, using seven different genotypes of pigeonpea and their hybrid progenies, revealed continuous variation for these traits. Hence, the type of gene action influencing in vitro cell proliferation and differentiation has been investigated in a diallel analysis of seven pigeonpea genotypes. Highly significant average heterosis was recorded for callus growth and seed yield/plant. In general, the F₁ hybrids which showed heterosis for callus growth also exceeded their better parent for yield/ plant. Combining ability analysis revealed both additive and non-additive gene effects for callus growth, while number of shoots/cotyledon was mostly governed by non-additive gene effects. The genotype, ICP 7035, was the best general combiner for callus growth and shoot forming capacity of cotyledons. Two cross combinations, 7186×6974 and 7035×T-21, showed maximum SCA effects for callus growth and shoots/cotyledon. Callus dry weight was positively correlated with seed yield/plant and seedling weight. The strong positive association of callus growth with seed yield indicates the possibility of using this system for mass screening and selection of superior hybrids.