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     

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).     

The importance of modelling heterogeneity in complex disease: application to NIMH Schizophrenia Genetics Initiative data

As for other complex diseases, linkage analyses of schizophrenia (SZ) have produced evidence for numerous chromosomal regions, with inconsistent results reported across studies. The presence of locus heterogeneity appears likely and may reduce the power of linkage analyses if homogeneity is assumed. In addition, when multiple heterogeneous datasets are pooled, inter-sample variation in the proportion of linked families () may diminish the power of the pooled sample to detect susceptibility loci, in spite of the larger sample size obtained. We compare the significance of linkage findings obtained using allele-sharing LOD scores (LOD_exp)—which assume homogeneity—and heterogeneity LOD scores (HLOD) in European American and African American NIMH SZ families. We also pool these two samples and evaluate the relative power of the LOD_exp and two different heterogeneity statistics. One of these (HLOD-P) estimates the heterogeneity parameter only in aggregate data, while the second (HLOD-S) determines separately for each sample. In separate and combined data, we show consistently improved performance of HLOD scores over LOD_exp. Notably, genome-wide significant evidence for linkage is obtained at chromosome 10p in the European American sample using a recessive HLOD score. When the two samples are combined, linkage at the 10p locus also achieves genome-wide significance under HLOD-S, but not HLOD-P. Using HLOD-S, improved evidence for linkage was also obtained for a previously reported region on chromosome 15q. In linkage analyses of complex disease, power may be maximised by routinely modelling locus heterogeneity within individual datasets, even when multiple datasets are combined to form larger samples.     

Quantitative studies on the mating system of opium poppy (Papaver somniferum L.)

Summary Nearly 400 individuals at two locations and over a number of years were crossed and subsequently scored for selfing versus outcrossing in eight monohybrid populations of opium poppy (Papaver somniferum). Two different marker loci, petal colour (R/r) and capsule size (B/b) were used to determine the male gametes that had effected fertilizations in F₂ recessives (rr and bb). The estimates of the outcrossing parameter were found to vary with year, location and for the marker locus used ( range: 0.0988–0.3704). Study of two dihybrid crosses involving the two loci simultaneously, further confirmed that outcrossing at the R/r locus was significantly greater than that at the B/b locus. The nature of the outcrossing was, in general, nonrandom. Selfmg predominated in this species; however, there was a high frequency of natural outcrossing for generating variations in P. somniferum.CIMAP publication No. 1086     

Quantitative studies on the mating system of jute (Corchorus olitorius L.)

Summary More than 100,000 individuals of C. olitorius were scored for selfing versus outcrossing in various populations, at several locations, over a number of years and seasons. Different marker loci, such as A ^d /a ⁰, Sh/sh, Cr/cr and Pl/pl, were used to determine the male gametes which had effected fertilization. The results showed that the frequency of outcrossing was extremely variable among loci, crosses and samples within a single locus. The outcrossing parameter, , was found to differ with years, locations and seasons within years. It was also found that outcrossing, in general, was nonrandom. Nonrandomness was also independent of flowering dates. The amount of outcrossing was directly associated with the frequency of F ₂ plants flowering at different dates. A recalculated outcrossing parameter from different authors' reported data, representing different years and locations, has been found to be nonrandom. It was observed that the propensity to outcross was not a simple function of changing gene frequency but was associated with the genotype of individual selected.     

Association of allozyme heterozygosity and sternopleural chaetae number inDrosophila melanogaster

Variation at two polymorphic allozyme loci (Adh and Gpdh) has been assayed in relation to sternopleural chaetae number in flies derived from a cage population ofD. melanogaster known to be under stabilising selection with respect to chaetae number. Central phenotypes were significantly more heterozygous at these loci than more extreme phenotypes. Analysis of the base popylation and of divergent directional selection lines derived from it suggested that alleles atAdh and Gpdh have a pleiotropic effect on chaetae number or are in strong linkage disequilibrium with chaetae factors. In addition, homozygotes atAdh and Gpdh were observed to have a higher variance in chaetae number than heterozygotes. It is concluded that additive and homeostatic models of gene action might both partly explain the higher heterozygosity of central phenotypes.     

Use of recombinant inbred lines of wheat for study of associations of high-molecular-weight glutenin subunit alleles to quantitative traits

Summary Recombinant inbred lines (RILs) derived by single plant descent to F₈ from a hybrid of Anza, a low-quality cultivar, and Cajeme 71, a high-quality cultivar, differed in alleles at three high-molecular-weight glutenin (HMW-glu) seed storage protein loci. The 48 RILs were classified by SDS-PAGE for the Anza alleles Glu-Alc (null), Glu-B1b (subunits 7 + 8), and Glu-D1a (subunits 2 + 12) and for Cajeme 71 alleles Glu-A1a (sub-unit 1), Glu-B1I (subunits 17 + 18), and Glu-D1d (subunits 5 + 10). All RILs and parents were grown in a replicated field trial with three levels of nitrogen (N) fertilization. Additive and additive x additive gene effects for the three loci were detected by orthogonal comparisons of means for each of six wheat end-use quality traits. Each HMW-glu genotype was represented by three to ten RILs so that variability among RILs within each HMW-glu genotype could be examined. N effects were consistently small. All traits except flour yield were highly correlated with predictor traits studied earlier. Flour protein content, baking water absorption, dough mixing time, bread loaf volume, and bread loaf crumb score were all correlated, suggesting similar gene control for these traits; however, specific additive locus contributions were evident: _B for flour yield; _B and _D for flour protein; and _B for absorption, but differing in sign; all three loci for mixing time, but _B was negative; and all three loci were positively associated with loaf volume. Digenic epistatic effects were significant for flour yield (_AD), flour protein (_AB), and absorption and mixing time (_AD, _BD). Only flour yield showed a trigenic epistatic effect. Six of seven epistatic effects were negative, thus showing how progress in breeding for high quality may be impeded by interaction of genes which, by themselves, have strong positive additive effects. Considerable genetic variance among RILs within a HMW-glu genotype was detected for all traits, and the summation of effects accounted for a mean of 13% of the parental differences for the six traits examined in this study. Clearly, further resolution of the genetics of wheat quality would be desirable from a plant breeding point of view.     

The Duffy blood group is linked to the α-spectrin locus in a large pedigree with autosomal dominant inheritance of Charcot-Marie-Tooth disease type 1

Summary The -spectrin locus (SPTA) on chromsome 1 maps to 1q22–q25 and -spectrin specific probes detect restriction fragment length polymorphisms (RFLPs) with the endonucleases MspI and PvuII. The Duffy blood group (FY) has been mapped to the 1p21–q23 region. We found positive linkage between the -spectrin and the Duffy loci with a maximal Lod score of 3.81 at =0.0 using the computer program MLINK. This indicates that both loci are very closely linked and probably localized to 1q22–q23.     

Contribution to the linkage theory of autopolyploids: I

A theory of linkage of autopolyploids is developed under consideration ofm loci andr alleles. The simplifying assumption of chromosome segregation, which may be considered as an approximation to the more adequate theory of chromatid segregation, is made throughout. Random mating and distinct, non-overlapping generations are assumed. Under these assumptions the problem is determined by three basic probability distributions—the distributions of genotypes and of gametes, and the segregation distribution. The segregation distribution is assumed to be the same for males and for females. The aim of the paper is to establish recurrence formulas (which allow to find the distributions of gametes and of genotypes from generation to generation, if the distribution of genotypes for an initial generation is known) and to investigate the limit behavior of these distributions as the number of generations increases indefinitely. In the present paper (hereafter referred to as I) the problem is explained, and the three characteristic distributions are introduced for the general case of a 2s-ploid,m loci, andr alleles. Recurrence relations are established for tetraploids,s=2 andm=2 loci, while the recurrence relations for the general case as well as the limit theorems will be given in the second part of this paper (hereafter referred to as II).     

Linkage studies on α-glycerophosphate and isocitrate dehydrogenases in Aedes albopictus (diptera: Culicidae)

Linkage studies were carried out on -glycerophosphate dehydrogenase (-GPDH) and isocitrate dehydrogenase (IDH) in the mosquito Aedes (Stegomyia) albopictus. Only one locus coding for -GPDH was revealed on agar gels by applying adult homogenates. Two loci for IDH were observed using either fourth-instar larvae, pupae, or adults. This study was restricted to the more anodal Idh-2 of the two loci, and -Gpdh. Both -Gpdh and Idh-2 encode dimeric enzymes. Thirteen backcrosses indicated that the -Gpdh and Idh-2 loci are arranged in linkage group 2 in the following order: p (pigmented pupa)—(ca. 2 map units)—Wb (white-body)—(7.5–17.8)—Idh-2—(13.1)—-Gpdh. Females exhibited more recombination than males.This work was supported by a Grant-in-Aid for Co-operative Research from the Ministry of Education, Japan.     

Evaluation of biometrical methods for estimating the number of genes

Summary Computer simulation of several genetic models was used to assess the effect of type I and type II statistical errors on estimating the number of genes by the inbred-backcross and genotype assay procedures. Depending upon the actual number of genes, heritability, and the probability of type I errors (), substantial upward and downward biases were observed in estimates of the number of genes from both methods. The estimated number of genes increased as was increased from 0.01 to 0.30 and as heritability increased. With high and/or high heritability, the estimated number of genes often exceeded the actual number. Downward biases occurred with low and low heritability, and tended to become greater as the number of genes in the model was increased. Large type II errors were associated with downward biases. The choice of had a greater impact on biases in estimates from the genotype assay procedure than from the inbred-backcross procedure. Increasing the number of backcrosses in the inbred-backcross procedure or delaying the assay generation in genotype assay increased the probability of upward biases in the estimated number of genes. Unbiased estimates can be obtained only by choice of an optimum . There is no known way to choose the optimum in practice. This fact reduces the value of estimates of the number of genes by genotype assay or by the inbred-backcross methods.Part of this study is based on the Ph.D. thesis of the senior author     

Linkage relationships in the bovine MHC region. High recombination frequency between class II subregions

Class II genes of the bovine major histocompatibility complex (MHC) have been investigated by Southern blot analysis using human DNA probes. Previous studies revealed the presence of bovine DO , DQ , DQ , DR and DR genes, and restriction fragment length polymorphisms for each of these genes were documented. In the present study, the presence of three additional class II genes, designated DZ , DY , and DY , are reported. DZ was assumed to correspond to the human DZ gene while the other two were designated DY because their relationship to human class II genes could not be firmly established. The linkage relationships among bovine class II genes and two additional loci, TCP1B and C4, were investigated by family segregation analysis and analysis of linkage disequilibrium. The results clearly indicated that all these loci belong to the same linkage group. This linkage group is divided into two subregions separated by a fairly high recombination frequency. One region includes the C4, DQ , DQ , DR and DR loci and the other one is composed of the DO DY , DY , and TCPIB loci. No recombinant was observed within any of these subregions and there was a strong or fairly strong linkage disequilibrium between loci within groups. In contrast, as many as five recombinants among three different families were detected in the interval between these subregions giving a recombination frequency estimate of 0.17 ± 0.07. The fairly high recombination frequency observed between class 11 genes in cattle is strikingly different from the corresponding recombination estimates in man and mouse. The finding implies either a much larger molecular distance between some of the bovine class II genes or alternatively the presence of a recombinational hot spot in the bovine class II region.     

Molecular and enzymatic properties of 1,2-α-d-mannosidase fromAspergillus saitoi

The molecular properties, such as molecular weight, N-and C-terminal amino acids, amino acid composition, and circular dichroism, of 1,2--mannosidase isolated from the culture filtrate ofAspergillus saitoi were determined.The enzyme had aK _m of 0.67 mM andk _cat of 1.27/s with mannobiose at pH 50.0 and 30°C. The anomeric configuration of the reaction products of the enzyme was examined by studying the -anomer. A single Manl2Man linkage in intact Taka-amylase A fromAspergillus oryzae was hydrolyzed, producing free mannose.     

Proximity of the CTLA-1 serine esterase and Tcr α loci in mouse and man

The serine esterase CTLA-1 gene was shown by in situ hybridization to map to the D segment of mouse chromosome 14, the same localization as a member of the immunoglobulin super family, Tcr . To further demonstrate the proximity of CTLA-1 and Tcr , genetic linkage was tested in mouse using restriction fragment length polymorphisms and a backcross progeny, and no recombination was observed in the 100 backcross products studied. Recombination events between Tcr /CTLA-1 and the markers Gdh-X and NP-1 show that the most probable order of these loci in the mouse 14D region is NP-1-Tcr /Ctla-1-Gdh-X. In man, the human homologue of CTLA-1 was shown by in situ hybridization to map on chromosome 14, at 14q11-q12, where Tcr also maps. Using the human cell line SUP-Tl, bearing the inversion inv(14)(q11;q32), we further demonstrated the loci order in man to be centromere-NP-1-Tcr -CTLA-1. To complement the cytogenetic and genetic mapping data, we tried to determine the physical distance between the two genes by pulsed field gel electrophoresis (PFGE). DNA prepared from various cell types, both mouse and human, were digested with a panel of rare cutter enzymes and hybridized first with CTLA-1, then with Tcr probes. None of the bands identified hybridized with both Tcr and CTLA-1 probes for either mouse or human cells. Although the physical mapping by PFGE is inconclusive, the cytogenetic and genetic data support close linkage of the Tcr and CTLA-1 genes in both mouse and man, suggesting homology between the D region of mouse chromosome 14 and the q11–q12 region of human chromosome 14, encompassing the Tcr and CTLA-1 loci. These findings also provide another example of proximity of genes coding for a member of the Ig super-family and a serine esterase.     

The biochemical genetics of rabbit erythrocyte esterases: Two new esterase loci

Starch gel electrophoresis of sonicated rabbit red cells using a modified system of buffers gave a maximum of 24 zones of esterase activity with -naphthyl acetate as substrate, 23 with -naphthyl propionate, and 20 with -naphthyl butyrate. Seventeen of the zones common to gels developed with any of these substrates were divided into three systems of isozymes, one of which, genetic system 1, has been previously described by Grunder et al. (1965). The zones of system 2 migrated between those of system 1 and those of system 3, the most anodal system. Each of these new systems, like system 1, consisted of three phenotypes controlled, respectively, by a pair of codominant autosomal alleles. In 10 test-cross matings involving systems 1 and 2, not more than two of four possible phenotypes were observed in the offspring of each family, thereby indicating close linkage of the two loci. Based on those matings, the probability that the two loci are independent was less than 0.0001.This work was supported in part by PHS Training Grant TI GM 701-07 and by the Serology Laboratory of the University of California at Davis.Adapted in part from a thesis by the senior author which was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of California at Davis, December 1968.     

Extragenic suppressor mutations of a β-tubulin mutation in the basidiomyceteCoprinus cinereus: Isolation and genetic and biochemical analyses

In total, 111 revertants were isolated from the oidia of a heat-sensitive, -tubulin mutant BEN193 (benA193) of the basidiomyceteCoprinus cinereus after mutagenesis by ultraviolet. Of the 111 revertants, 48 were genetically analyzed. In 15 of the 48 revertants, reversion was due to mutations at loci unlinked tobenA, whereas in the remaining 33 revertants, reversion was due to mutations withinbenA or at loci closely linked tobenA. The 15 extragenic suppressor mutations comprised three groups in terms of genetic linkage; two of them were designated asmipA andmipB. Suppressor mutations in the third group were found to bebenC, one of the four loci we have previously identified as genes conferring benomyl resistance. Biochemical analysis revealed thatbenC ⁺ is a structural gene for a major -tubulin in vegetative hyphae.     

Mating system and multilocus associations in a natural population of Pseudotsuga menziesii (Mirb.) Franco

Summary Arrays of open-pollinated seeds were assayed for allozyme polymorphisms at ten loci (Aat2, Est1, G6pd, Idh, Mdh2, Mdh3, Pgm, Sod, 6Pgd1, 6Pgd2) to obtain estimates of the outcrossing rate and assess multilocus association in a natural population of coastal Douglas-fir, Pseudotsuga menziesii (Mirb.) Franco. The allele frequencies in the samples of adult trees and pollen-gamete pool were similar. Maximum-likelihood estimators of the outcrossing rate for individual loci and two multilocus models were derived using counting methods. The single-locus maximum likelihood estimates (MLEs) of the outcrossing rate were significantly heterogeneous; they varied over a more than two-fold range from 0.404 to 0.935, with an average MLE of 0.741. Both multilocus MLEs of the outcrossing rate were 0.887. The sample of trees was in random mating equilibrium when assessed on a pairwise-locus basis using Burrows' composite measure of gametic disequilibrium, with one exception (Mdh2 Sod) that was attributable to a rare gametic class. In the sample of pollen gametes, 5 of the 45 pairwise-locus associations were nominally significant at the 0.05 level: Idh Est1, Mdh2 Sod, Aat2 Est1, Aat2 Mdh3, and Est1 Mdh3. These apparent associations were attributable in most cases to the relative excess of uncommon or rare paternal gametes of discernibly outcrossed embryos. An additional two-locus association was identified for Mdh2 Pgm which was marginally significant for the major partition of the contingency table that excluded paternal gametes with the rare allele Mdh2 ² .     

Segregation,linkage, and diversity of allozymes in knobcone pine

Summary Female gametophytes of knobcone pine were used to study genetic variation at 58 loci in 26 enzyme systems. Mendelian segregation and linkage were tested at 21 loci. Got1, Pgi2, Mnr3, Adh2, and Lap2 were linearly arrayed in a single linkage group. Est and Acp3, and Flest and Lap1, formed two independent linkage groups. Although Mendelian segregation was the rule, several cases of segregation distortion were observed. Pooled over trees, Lap1 and Aap1 showed significant distortion. Of 11 cases of distortion observed for individual trees, 10 showed an excess of common alleles. Pooled over both loci and trees, giving a total sample of 17,183 gametes, the common alleles were significantly overrepresented by 1.1%, and heterogeneity was highly significant. Our results, and others in the literature, suggest that segregation distortion may affect the genetic structure of conifer populations.     

The hemoglobins of Artemia salina. V. Genetic variation in hemoglobin 1, hemoglobin 2, and hemoglobin 3

Electrophoretic mobilities of three hemoglobins (Hb1, Hb2, and Hb3) were studied in 15 populations of brine shrimps. Genetic segregation data support the model that Hb2 contains n -polypeptides and n -polypeptides; Hb1 contains 2n -polypeptides. Hb3 contains neither - nor -polypeptides. There is no evidence of linkage of and loci with each other or with the locus (or loci) which governs Hb3 or with the nonhomologous portion of the sex chromosomes. Hemoglobins of different populations may be hybridized in vitro by incubation at high temperature. Reversible dissociation to subunits which contain only one ( or ) polypeptide occurs at 40 C (for Hb1) and at 50 C (for Hb2).Supported by Grant HD-11445 from the National Institutes of Health.     

A linkage map ofBrassica rapa (syn.campestris) based on restriction fragment length polymorphism loci

Summary A detailed linkage map ofB. rapa (syn.campestris) was constructed based on segregation of 280 restriction fragment length polymorphism loci, detected by using 188 genomic DNA clones as probes on DNAs from a F₂ population of Chinese cabbage MichihilF×Spring broccoli. These genetic markers covered 1,850 centiMorgans (cM) and defined ten linkage groups, which equals the haploid chromosome number of this species. Extensive sequence duplication was evident by the detection of two or more segregating loci with each of 69 clones (36.7% of the total). Although some duplicated loci were randomly distributed throughout the genome, many had linkage arrangements that were conserved on different linkage groups, suggesting that large chromosome fragments were present in multiple copies. However, conservation in the linkage arrangement of duplicate loci throughout entire pairs of linkage groups was not observed. Single-copy loci were often found to be located within conserved duplicated regions, and linkage distances between some loci having conserved duplicated arrangements were substantially different between the duplicated regions. Structural rearrangements, such as insertions, deletions, and inversions or combinations of these events, seemed to be related to the alternations of map distances between duplicated loci and to the dispersal of duplicated chromosome fragments. These results suggest thatB. rapa has evolved in part by duplication of chromosomes or large chromosome fragments with subsequent structural rearrangements.