Number Of Quantitative Traits Research Articles

Morphological and molecular characterization of variation in common bean (Phaseolus vulgaris L.) germplasm from Azad Jammu and Kashmir, Pakistan.

Phaseolus vulgaris, an essential food and source of protein, is cultivated across the world. This study was carried out to investigate the diversity and population structure of 34 P. vulgaris landrace accessions collected from the Azad Jammu and Kashmir (AJ&K) regions of Pakistan. The samples were analyzed both morphologically and using genetic variation identified through RNA sequencing. Our results indicated that most genetic variation occurs among local accessions, with little genetic variation occurring between geographical regions. In addition, the accessions fell into two major genetic groups. Morphological analysis revealed that these two genetic groups differ in a number of quantitative traits, including seed length, seed width, and seed weight. One accession, DUD-11, appears to be a mixture of the two major groups genetically as well as morphologically. Among the other accessions, DUD-8, RWK-2, and NGD-1 depicted particularly high seed weight along with higher seed length, seed width, and seed yield per plant. We suggest focusing on these accessions in future breeding programs. More generally, our results provide baseline data that will be useful for crop improvement and effective cultivation practices in Pakistan.

Comparison of the F2 and F3 traits’ inheritance by the rice hybrid ‘Kontro x Kuboyar’

Rice is a very moisture-loving crop, therefore it suffers greatly from drought. However, in the world in recent years, there is an increasing shortage of fresh water, which makes the use of upland rice relevant. Therefore, rice varieties with a reduced demand for water supply, developed as a result of breeding work, are more profitable for agricultural producers than agrotechnical or reclamation measures. It is much cheaper to produce such rice with periodic irrigation. The purpose of the current study was to analyze genetically the main quantitative traits of the rice hybrid ‘Kontro x Kuboyar’ with the selection of the best forms for subsequent breeding for drought resistance. The study was carried out in 2019–2020 on the rice plots of the special subdivision ‘Proletarskoye’ belonging to of the Agricultural Research Center “Donskoy” in the Rostov Region. In the course of the study there has been made a genetic analysis of a number of quantitative traits that determine rice productivity. The character of traits’ inheritance in the second and third generations was identical. The inheritance of the trait ‘panicle length’ was characterized by partial negative dominance and monogenic differences in parental varieties. The trait ‘number of spikelets per panicle’ was inherited according to the type of negative dominance of smaller values of the trait and was determined by the interaction of two pairs of genes. The trait ‘1000-kernel weight’ was identified by the partial dominance of large values of the trait and monogenic differences in the parental varieties. The traits ‘kernel length’ and ‘kernel width’ were split according to the monohybrid scheme 1:2:1, and there was no dominance. From the third generation, the best early-maturing forms with optimal values of the traits ‘plant height’, ‘panicle length’, ‘1000-kernel weight’ and ‘the best kernel percentage per panicle’ were selected for the subsequent breeding of drought-resistant rice varieties.

The effect of a number of quantitative traits on seed productivity of alfalfa samples

The current paper has presented the correlation between alfalfa seed productivity and the values of morphological characteristics for a three-year growing cycle. The purpose of the study was to estimate the correlation between the alfalfa samples' productivity and the elements of its structure and other traits in order to find their optimal values. The study was carried out in 2017-2019 on the experimental plots of the Agricultural Research Center “Donskoy”. The material for the research was a working collection of 105 alfalfa samples, varieties and lines, and the standard variety ‘Rostovskaya 90'. The crops were sown in row, in double repetition, with row spacing of 20 cm on a plot of 1m2, the seeding rate was 2 g/m2. The seed productivity of the collection samples varied from 27.2 g/m2 (the sample ‘SGP-128') to 101.0 g/m2 (the sample ‘SGP-414'). The average productivity of the standard variety ‘Rostovskaya 90' was 53.2 g/m2. About 23% of the samples that exceeded the standard variety produced 60-101 g/m2. Analysis of the graphs of the correlation between productivity and other traits made it possible to determine their optimal values, namely height of alfalfa plants varied from 80.3 (‘SPCh-434') to 107.3 cm (‘SGP-414'). The optimum plant height was between 94 and 98 cm. The number of beans per bunch in the studied samples varied from 8.8 to 16.2 pcs.; the optimal number of beans per bunch was from 12 to 13 pcs.; the number of turns on a bean was from 1.4 to 2.8, the optimal number was 2.4-2.6. The number of seeds in a bean varied on average from 3.1 to 6.4 pcs., the optimal number was 5.0-5.5 pcs. These values have formed the basis for the model of the alfalfa variety. The study of the alfalfa collection resulted in the identification several samples due to their productive and economically valuable traits.

The analysis of trait inheritance of rice hybrid from the crossing of the upland rice sample “Kontro” with the productive rice variety “Kuboyar”

Rice is an important agricultural crop. It can grow not only when the flooded soil, but in the conditions of periodic irrigation, both artificial and natural (rain). The production of such rice is much cheaper. In addition, there is a shortage of fresh water in the world, which makes the use of upland rice relevant. The purpose of the current work was to develop the productive drought-resistant rice lines based on the “Kontro x Kuboyar” hybrid. The main tasks were to hybridize the varieties, to analyze genetically a number of quantitative traits and to select the best plants that could form compact full-grained panicles for further breeding work and testing for drought resistance. There has been conducted a genetic analysis of varying quantitative traits affecting the rice productivity. The parental forms did not differ in plant height. Along the panicle length there was a partial negative dominance and monogenic differences between the hybridized varieties. According to the number of spikelets per panicle there has been identified a negative dominance of lower values and the interaction of two pairs of genes. According to 1000 grain weight, there was a partial dominance of large values of the trait and monogenic differences in the original parental forms. Splitting along the length and width of the caryopsis was in a ratio of 1 : 2 : 1, which indicated monogenic differences between the parental varieties. There have been selected the best recombinant rice forms, combining a compact panicle with a large number of grains, and there has been developed the initial material for practical breeding. The study was carried out in 2019 on the plots of the Separate Subdivision “Proletarskoye” of the FSBSI “ARC “Donskoy” in the Rostov Region.

Inheritance of quantitative traits of the rice hybrid F<sub>2</sub>-F<sub>3</sub> Svetly × Mavr

Rice can form not only white, but also red, brown and black kernels. In black rice, pericarp kernels contain anthocyanin, which has an antioxidant effect and is beneficial to human health. The current paper is devoted to the development of new rice lines with black pericarp based on the hybrid Svetly × Mavr. The purpose of the work is to develop new forms of rice with anthocyanin pericarp by a hybridization of varieties with white and black kernels, a genetic analysis of a number of quantitative traits and by a selection of the best variants with compact vertical black-kernel panicles to use them in future breeding work. There has been conducted a genetic analysis of varying quantitative traits affecting the rice productivity. The cleavage by a pericarp color was carried out according to the type of recessive epistasis in a ratio of 9 : 3 : 4. The black color of the pericarp was formed in the presence of two dominant genes Pb and Pp in the genotype, the brown color was due to the Pb gene, the white color was due to the rest combinations of genes. The traits “plant height” and “number of spikelets per panicle” of the hybrids F2 and F3 were inherited by the type of superdomination of large values of the trait controlled by two loci, the interaction of which resulted in transgressive forms. According to the traits “panicle length” and “1000 kernel weight”, there has been identified intermediate monogenic inheritance and two-peaked characteristic frequency distribution curves. There have been identified rice forms with black pericarp color, combining the optimal values of the traits “plant height”, increased “panicle length”, “number of spikelets per panicle” and “1000 kernel weight”. On its basis there has been also developed an initial material for practical breeding. The study was carried out in 2018–2020 on the fields of the EP “Proletarskoye” of the Rostov region.

Fast Multi-Task SCCA Learning with Feature Selection for Multi-Modal Brain Imaging Genetics.

Brain imaging genetics studies the genetic basis of brain structures and functions via integrating both genotypic data such as single nucleotide polymorphism (SNP) and imaging quantitative traits (QTs). In this area, both multi-task learning (MTL) and sparse canonical correlation analysis (SCCA) methods are widely used since they are superior to those independent and pairwise univariate analyses. MTL methods generally incorporate a few of QTs and are not designed for feature selection from a large number of QTs; while existing SCCA methods typically employ only one modality of QTs to study its association with SNPs. Both MTL and SCCA encounter computational challenges as the number of SNPs increases. In this paper, combining the merits of MTL and SCCA, we propose a novel multi-task SCCA (MTSCCA) learning framework to identify bi-multivariate associations between SNPs and multi-modal imaging QTs. MTSCCA could make use of the complementary information carried by different imaging modalities. Using the G 2,1-norm regularization, MTSCCA treats all SNPs in the same group together to enforce sparsity at the group level. The -norm penalty is used to jointly select features across multiple tasks for SNPs, and across multiple modalities for QTs. A fast optimization algorithm is proposed using the grouping information of SNPs. Compared with conventional SCCA methods, MTSCCA obtains improved performance regarding both correlation coefficients and canonical weights patterns. In addition, our method runs very fast and is easy-to-implement, and thus could provide a powerful tool for genome-wide brain-wide imaging genetic studies.

Breeding of the Russian sable: Stages of industrial domestication and genetic variability

Creating farms for sable breeding was associated with the commercial destruction of natural populations and, consequently, the overall decline in the species number. The gene pool of the first farm-bred sable population in Russia, established in the vicinity of Moscow (“Pushkinskiy” fur farm), was formed by crossing of animals removed from nine natural populations. In the first eight years of farm operation, approximately one thousand animals were used for sable breeding; some of these animals were able to adapt to the farm management and, subsequently, to the selection for a number of quantitative traits in the period of industrial domestication. It took about ten years for breeders to work out the breeding and selection technologies, which became successfully employed in the established affiliated sable breeding farms. The main achievement in sable breeding over the 85-year historical period of breeding in Russia is the creation of two unique breeds, black sable (1969) and Saltykovskaya 1 (2007). In general, industrial domestication in fur farming and the subsequent breeding works made the fur of many species (mink, fox, Arctic fox) obtained from natural populations uncompetitive, which undoubtedly reduced the hunting interest in the animals living in the wild. Consequently, hunting for fur-bearing animals of most species decreased and has only local importance. Owing to the specific features of sable biology, the fur of farm-bred animals cannot yet completely replace the furs obtained by hunting; however, the farm-bred sable population is constantly growing. This review presents the results of the analysis of the level of genetic variability in natural and farm populations at nuclear and mitochondrial loci. The comparative analysis makes it possible to estimate the loss of genetic diversity upon the species adaptation to the new conditions of existence.

Assessment of trait efficiency and selection of parents in brinjal (<i>Solanum melongena</i> L.)

Twenty one brinjal genotypes with a number of quantitative traits were assessed under biometric analysis for evaluating efficient agro-economic traits and selecting promising parents. No. of fruits in inflorescence/plant, fruits/plant, yield/plant (g), fresh weight/fruit (g), harvesting period and no. of fruits in solitary/plant can consider as favorable attributes for improvement through selection because of their higher GCV, PCV, heritability and genetic advance as percent of mean. Emphasis can be given to characters with high positive direct effect on yield like no. of fruits in inflorescence/plant, fresh weight/fruit (g) and no. of secondary branches/plant. Improvement would be more efficient with selection index I 134 based on three characters (yield/plant (g) + no. of fruits in inflorescence/plant + no. of fruits in solitary/plant) which showed genetic gain 3.39 and relative efficiency 258.14% significantly higher to other indexes based on four or more characters. Biplot analysis using selection score and decisive trait index indicated that Sada Begun, Debjhuri Hajari, Kajla, and BARI-9 can be the promising parents for future breeding programme.

The influence of particular chromosome regions of Triticum timopheevii on the formation of resistance to diseases and quantitative traits in common wheat

Evaluation of the influence of Triticum timopheevii Zhuk. introgression fragments (2n = 28, AtAtGG) and their combinations on resistance to leaf and stem rust, powdery mildew, and a number of quantitative traits in 15 introgressive lines of common wheat was conducted. Analysis of introgressive lines by molecular genetic and cytological methods showed the efficiency of using a complex of different types of markers for detailed characterization of hybrid forms and detection of different translocations and substitutions. Evaluation of lines according to their resistance to fungal diseases showed that lines that contain an introgressive fragment of the 5G chromosome in their genome are completely resistant to populations of leaf rust in Western Siberia and the stem rust that is typical for Omsk oblast. Lines 3862-5 and 3862-15, which contain a fragment of the long arm of the 2G chromosome in their genome, were resistant to the population of stem rust in Western Siberia. Introgressive lines were studied for a number of quantitative traits. No negative influence of alien material on the yield and other quantitative traits was observed in all studied lines, which allows one to use them in breeding as donors of resistance to fungal diseases. In addition, the positive influence of T. timopheevii 2G chromosome fragments on the number of grains in an ear was determined.

Molecular systematics of the parasitic genus Conopholis (Orobanchaceae) inferred from plastid and nuclear sequences

Little is known of the evolutionary relationships within Conopholis, a small holoparasitic genus belonging to the broomrape family. Presently, Conopholis is described as having two species, C. americana and C. alpina. This classification is based on a combination of presence/absence of morphological characters along with a number of quantitative traits. We assessed the relationships among populations and species of this genus to determine whether the present taxonomic hypothesis is reflected in molecular phylogenies. We conducted the first phylogenetic study of Conopholis using plastid (trnfM-E intergenic spacer and clpP gene/introns) and nuclear (PHYA intron 1) sequences from a wide taxonomic sampling covering its entire geographical range in North America. Analyses were carried out using a variety of phylogenetic inference approaches. Reciprocal monophyly between the two traditionally accepted species has not yet been achieved. Instead, three distinct genetic clusters were recovered. Conopholis alpina is clearly paraphyletic and shows evidence of belonging to at least two distinct lineages. Specimens found in Costa Rica and Panama form a distinct group from those located in northern Mexico and the southwestern United States. The monophyly of C. americana was also not recovered; however, the possibility of it being monophyletic could not be rejected with confidence. These analyses recovered three distinct lineages indicating that there could be a minimum of three species within the genus. A reevaluation of morphological features within Conopholis may reveal shared features that could further corroborate our molecular findings.

CORRELATION , PATH COEFFICIENTS AND REGRESSION ANALYSIS IN SUMMER SQUASH.

Simple correlation coefficients were estimated among number of quantitative traits from an experimental data applied by RCB Design with three replications, involved two squash cultivars (Mulla-Ahmed and the hybrid Opal type-HED 103) and four different concentrations of K and IAA (0.0 , 0.5 , 1.0 , 1.5 gl-1) , (0.0 , 100 , 200 , 300 mgl-1) respectively. In order to determine the best traits affecting on the yield, path coefficient analysis was used to divide the correlation coefficients of the yield with its components into direct and indirect effects. To determine the nature response of the different traits for both K and IAA, trend analysis was applied, in which mean squares were divided in a variance analysis table into linear, quadratic and cubic relationship. Significant correlation coefficients in a positive direction were obtained between the total yield and its contributing characters viz. (no . of female flowers , sex ratio , no. of fruits plant -1 , fruit weight and early yield). Number of fruits plant -1 was characterized with highest direct effect on the total yield , whereas, the highest indirect effects resulted from no. of fruits through fruit weight and from no. of female flowers through the path of no. of fruits plant -1. Trend analysis revealed a significant quadratic order regression form between K concentrations with plant height and fruit weight , while the relation between IAA concentrations with no. of branches and early yield showed a significant effect at the second order equation form.

PROTECTED POLYMORPHISM AND EVOLUTIONARY STABILITY IN PLEIOTROPIC MODELS WITH TRAIT-SPECIFIC DOMINANCE

When alleles have pleiotropic effects on a number of quantitative traits, the degree of dominance between a pair of alleles can be different for each trait. Such trait-specific dominance has been studied previously in models for the maintenance of genetic variation by antagonistic effects of an allele on two fitness components. By generalizing these models to an arbitrary number of fitness components or other phenotypic traits with different degrees of dominance, I show that genetic polymorphism is generally impossible without antagonistic fitness effects of different traits and without trait-specific dominance. I also investigate dominance and pleiotropy from a more long-term evolutionary perspective, allowing for the study of general ecological scenarios, and I discuss the effects of trait-specific dominance on evolutionary stability criteria. When selection is mainly directional and only trait-specific dominance and antagonism cause the emergence of polymorphism, then these polymorphisms can be overtaken by single mutants again, such that they are probably short-lived on an evolutionary time scale. Near evolutionarily singular points where directional selection is absent, trait-specific dominance and overdominance facilitate the emergence of polymorphism and cause evolutionary divergence in some cases. An important outcome of these models is that trait-specific dominance allows for the emergence of genetic polymorphisms without a selective disadvantage for heterozygotes. This removes the scope for the evolution of assortative mate choice and affects dominance modification. Sympatric speciation by disruptive ecological selection requires this heterozygote disadvantage in order to evolve, and therefore it becomes less plausible if the emergence of genetic polymorphism usually occurs via trait-specific dominance and antagonistic effects.

Mean curvature versus normality: A comparison of two approximations of Fisher's geometrical model

Fisher's geometrical model amounts to a description of mutation and selection for individuals characterised by a number of quantitative traits. In the present work the fitness landscape is not assumed to be spherically symmetric, hence different points, i.e. phenotypes, on a surface of constant fitness generally have different curvatures. We investigate two different approximations of Fisher's geometrical model that have appeared in the literature. One approximation uses the average curvature of the fitness surface at the parental phenotype. The other approach is based on a normal approximation of a distribution associated with new mutations. Analytical results and simulations are used to compare the accuracy of the two approximations.

The frequency of the perfect genotype in a population subject to pleiotropic mutation

We consider a large population of asexual organisms characterised by a number of quantitative traits that are subject to stabilising selection. Mutation is taken to act pleiotropically, with every mutation generally changing all of the traits under selection. We focus on the equilibrium distribution of the population, where mutation and selection are in balance. It has been previously established that the equilibrium distribution of genotypic effects may be anomalous, as it may contain a singular spike—a Dirac delta function—corresponding to a non-zero proportion of the population having exactly optimal genotypic values. In the present work, we present exact results for the case where three traits are under selection. These results give the equilibrium genetic variance of the population, and the proportion of the population that have the optimal genotype. This is achieved for two different spherically symmetric distributions of mutant effects. Additionally, a simple and robust numerical approach is also presented that allows the treatment of some other mutation distributions, where there are an arbitrary number of selected traits.

Effect of a substitution of a short chromosome segment carrying a histone H1 locus on expression of the homeotic gene Tl in heterozygote in the garden pea Pisum sativum L.

Mutation tlx was induced in pea by gamma-irradiation in the homeotic gene Tl, responsible for transformation of the upper leaflets into tendrils. Heterozygotes tlx/Tl+ possess ‘flat tendrils’ having narrow rudimentary leaf blades. By 15 generations of selfing the line Delta was established, in which closely linked loci Tl and His1 were maintained heterozygous, His1 encoding the most abundant histone H1 subtype. The heterozygous chromosome region was flanked by several recessive markers. The genotype of line Delta with respect to loci His1 and Tl was: tlxHis12/Tl+His11. There were derived two sublines, Delta-s and Delta-f, homozygous for each of the His1 alleles but heterozygous for the Tl and flanking markers. The three lines were compared for a number of quantitative traits, including parameters of flat tendrils reflecting the expression of the gene Tl in heterozygote, and for recombination rate within the heterozygous segment. The line Delta-s exceeded Delta-f in terms of the level of the first inflorescence, length of the internode 8–9, number of pods on the main stem and mean seed mass, but was inferior regarding the mean number of seeds per pod. The flat tendrils were wider in the Delta-s as compared with the Delta-f line ; the difference was highly significant and exceeded 15% at node 17. Recombination rate between the flanking markers also showed significant differences, being highest in the Delta-f and lowest in the Delta line. We do not exclude the possibility that the observed differences in the quantitative traits are caused by the substitution of alleles of the histone H1 gene. The effects of such substitution appeared to be especially strong for expression of the homeotic gene Tl.

Comment – Genes and temperament, a shortcut for unravelling the genetics of psychopathology?

Personality is complex and unique, in that people differ greatly from one another in multiple components of personality. Nevertheless, personality can be measured and dissected into a number of quantitative traits. One of the more recent tools is the Temperament and Character Inventory (TCI), which is a self-report personality questionnaire based on Cloninger's psychobiological model of personality (Cloninger et al., 1994). This model consists of four dimensions of temperament and three dimensions of character. Temperament dimensions (novelty-seeking, harm avoidance, reward dependence and persistence) refer to automatic emotional responses to stimuli and are moderately heritable, relatively stable throughout life, and invariant despite sociocultural influences (Svrakic et al., 1996). Individuals high in novelty-seeking tend to be quick-tempered, excitable, exploratory, curious, enthusiastic, impulsive and disorderly. Individuals high in harm avoidance tend to be cautious, careful, fearful, tense, apprehensive, nervous, discouraged, insecure, negativistic and pessimistic, even in situations which do not worry other people. People scoring high on reward dependence seek social contact and are open to communication with other people; they are tender-hearted, sensitive, dedicated and sociable. Highly persistent individuals tend to be industrious, hard-working, persistent, and stable, despite frustration and fatigue. The character dimensions (self- directedness, cooperativeness and self-transcendence) involve differences in social goals and values and seem to be moderately influenced by family environment (Cloninger et al., 1994). Cloninger's psychobiological model of personality shows correlations to other factor models of personality, like the NEO five-factor model. The TCI dimension of novelty-seeking shows positive loading on NEO conscientiousness (Cloninger et al., 1994).

The contribution of quantitative trait loci and neutral marker loci to the genetic variances and covariances among quantitative traits in random mating populations.

Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.

Phenotypic effect of substitutions of short chromosomal segments containing different alleles of histone HI genes in garden pea (Pisum sativum L.)

SummaryA hypothesis has been tested that alterations in the molecule of histone H1 are capable of influencing quantitative traits of an organism. Two pairs of isogenic lines were constructed by selfing of plants kept heterozygous either for allelic variants of histone H1 subtype 1 (His1 gene) or for allelic combinations (haplotypes) of closely linked genes of H1 subtypes 3, 4, 5 and 6 (gene clusterHis(2–6)). After 19 and 15 generations of selfing, respectively, expectation of the length of chromosome which remained heterozygous has comprised 2·6 cM nearHisl and 3·5 cM nearHis(2–6). The third pair of isogenic lines was obtained as a result of two successive intracluster cross-over events bordering the gene of the subtype 5 (His5). In each pair of isogenic lines there have been revealed some statistically significant differences between mean values of a number of quantitative traits.

ANALYSIS OF THE NUCLEAR GENOME, AND CONSTRUCTION OF A GENETIC MAP FOR PEACH [Prunus persica (L.) BATSCH.

Flow cytometric analysis, of leaf nuclei from three cultivars, was used to estimate the DNA content of peach (∼0.61 pg or ∼0.59 × 109 bp/diploid nucleus; 2x=16), and ndicated that the peach genome is only slightly larger than that of Arabidopsis. This value was indirectly confirmed by measurements of nuclei from haploid, triploid and “tetraploid” (cytochimera) peach accessions. cDNA and genomic clones have been used to determine the level of polymorphism among various peach cultivars and related species. Overall, ∼33% of the clones detected polymorphic loci. As expected, the highest level of polymorphism was found in interspecific hybrids (∼50%); whereas in intraspecific populations, only 1 in 5 genomic clones, and 1 in 3 cDNA clones were able to detect polymorphisms (RFLPs). These clones, as well as RAPD primers, are being used to construct a genetic linkage map by analyzing their segregation in 3 intraspecific peach populations (an Fl from France and two F2s from the U.S.). Taken together, these populations are segregating for 12 Mendelian traits and a number of quantitative traits. Our results have enabled us to identify a number of linkage groups, some composed of both molecular and phenotypic markers. The current structure of the peach map is reported.

Second-order approximations for selection coefficients at polygenic loci. II. Pleiotropy.

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). I demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.