Genetic variance, coefficient of parentage, and genetic distance of six soybean populations

Plant breeders would like to predict which biparental populations will have the largest genetic variance. If the population genetic variance could be predicted using coefficient of parentage or genetic distance estimates based on molecular marker data, breeders could choose parents that produced segregating populations with a large genetic variance. Three biparental soybean {Glycine max (L.) Merr.} populations were developed by crossing parents that were closely related, based on pedigree relationships. Three additional biparental populations were developed by crossing parents that were assumed to be unrelated. The genetic variance of each population was estimated for yield, lodging, physiological maturity, and plant height. Coefficient of parentage was calculated for each pair of parents used to develop the segregating populations. Genetic distance was determined, based on the number of random amplified polymorphic markers (RAPD) that were polymorphic for each pair of parents. Genetic distance was not associated with the coefficient of parentage or the magnitude of the genetic variance. The genetic variance pooled across the three closely related populations was smaller than the genetic variance pooled across the three populations derived from crossing unrelated parents for all four traits that were evaluated. Received: 24 April 1996 / Accepted: 17 May 1996     

The use and abuse of genetic marker‐based estimates of relatedness and inbreeding

Genetic marker‐based estimators remain a popular tool for measuring relatedness (r_xy) and inbreeding (F) coefficients at both the population and individual level. The performance of these estimators fluctuates with the number and variability of markers available, and the relatedness composition and demographic history of a population. Several methods are available to evaluate the reliability of the estimates of r_xy and F, some of which are implemented in the program COANCESTRY. I used the simulation module in COANCESTRY since assess the performance of marker‐based estimators of r_xy and F in a species with very low genetic diversity, New Zealand's little spotted kiwi (Apteryx owenii). I also conducted a review of published papers that have used COANCESTRY as its release to assess whether and how the reliability of the estimates of r_xy and F produced by genetic markers are being measured and reported in published studies. My simulation results show that even when the correlation between true (simulated) and estimated r_xy or F is relatively high (Pearson's r = 0.66–0.72 and 0.81–0.85, respectively) the imprecision of the estimates renders them highly unreliable on an individual basis. The literature review demonstrates that the majority of studies do not report the reliability of marker‐based estimates of r_xy and F. There is currently no standard practice for selecting the best estimator for a given data set or reporting an estimator's performance. This could lead to experimental results being interpreted out of context and render the robustness of conclusions based on measures of r_xy and F debatable.     

Specifying the introgressed regions from H. argophyllus in cultivated sunflower (Helianthus annuus L.) to mark Phomopsis resistance genes

A method based upon targetting of intro-gressed markers in a Phomopsis-resistant line (R) of cultivated sunflower, issuing from a H. argophyllus cross was used to mark the Phomopsis resistance regions. Our study was based upon 203 families derived from a cross between an inbred line susceptible to Phomopsis (S1) and the introgressed resistant line (R). Families were checked for Phomopsis resistance level in a design with replicated plots and natural infection was re-inforced by pieces of contaminated stems. Thirty four primers were employed for RAPD analysis. Out of 102 polymorphic fragments between (S1) and H. argophyllus, seven were still present in (R) suggesting that they marked introgressions of H. argophyllus into (R). The plants were scored for the presence or absence of 19 fragments obtained from five primers, and the relationships between the presence/absence of fragments in plants and Phomopsis resistance/susceptiblity in the progenies was determined by using an analysis of variance. We found that at least two introgressed regions, as well as favourable factors from sunflower, contributed to the level of Phomopsis resistance in cultivated sunflower. Received: 28 June 1996 / Accepted: 5 July 1996     

MAXIMUM-LIKELIHOOD APPROACHES APPLIED TO QUANTITATIVE GENETICS OF NATURAL POPULATIONS

Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least-squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F-ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within-generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative-genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.     

Shrinkage Estimators for Covariance Matrices

Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.     

Gliadin polymorphism in wild and cultivated einkorn wheats

To study the relationships between different species of the Einkorn group, 408 accessions of Triticum monococcum, T. boeoticum, T. boeoticum ssp. thauodar and T. urartu were analyzed electrophoretically for their protein composition at the Gli-1 and Gli-2 loci. In all the species the range of allelic variation at the loci examined is remarkable. The gliadin patterns of T. monococcum and T. boeoticum were very similar to one another but differed substantially from those of T. urartu. Several accessions of T. boeoticum and T. monococcum were shown to share the same alleles at the Gli-1 and Gli-2 loci, confirming the recent nomenclature that considers these wheats as different subspecies of the same species, T. monococcum. The gliadin composition of T. urartu resembled that of the A genome of polyploid wheats more than did T. boeoticum or T. monococcum, supporting the hypothesis that T. urartu, rather than T. boeoticum, is the donor of the A genome in cultivated wheats. Because of their high degree of polymorphism the gliadin markers may help in selecting breeding parents from diploid wheat germ plasm collections and can be used both to search for valuable genes linked to the gliadin-coding loci and to monitor the transfer of alien genes into cultivated polyploid wheats. Received: 8 July 1996 / Accepted: 12 July 1996     

Phylogenetic Depth of the Bacterial Genera Aquifex and Thermotoga Inferred from Analysis of Ribosomal Protein, Elongation Factor, and RNA Polymerase Subunit Sequences

The phylogenetic placement of the Aquifex and Thermotoga lineages has been inferred from (i) the concatenated ribosomal proteins S10, L3, L4, L23, L2, S19, L22, and S3 encoded in the S10 operon (833 aa positions); (ii) the joint sequences of the elongation factors Tu(1α) and G(2) coded by the str operon tuf and fus genes (733 aa positions); and (iii) the joint RNA polymerase β- and β′-type subunits encoded in the rpoBC operon (1130 aa positions). Phylogenies of r-protein and EF sequences support with moderate (r-proteins) to high statistical confidence (EFs) the placement of the two hyperthermophiles at the base of the bacterial clade in agreement with phylogenies of rRNA sequences. In the more robust EF-based phylogenies, the branching of Aquifex and Thermotoga below the successive bacterial lineages is given at bootstrap proportions of 82% (maximum likelihood; ML) and 85% (maximum parsimony; MP), in contrast to the trees inferred from the separate EF-Tu(1α) and EF-G(2) data sets, which lack both resolution and statistical robustness. In the EF analysis MP outperforms ML in discriminating (at the 0.05 level) trees having A. pyrophilus and T. maritima as the most basal lineages from competing alternatives that have (i) mesophiles, or the Thermus genus, as the deepest bacterial radiation and (ii) a monophyletic A. pyrophilus–T. maritima cluster situated at the base of the bacterial clade. RNAP-based phylogenies are equivocal with respect to the Aquifex and Thermotoga placements. The two hyperthermophiles fall basal to all other bacterial phyla when potential artifacts contributed by the compositionally biased and fast-evolving Mycoplasma genitalium and Mycoplasma pneumoniae sequences are eschewed. However, the branching order of the phyla is tenuously supported in ML trees inferred by the exhaustive search method and is unresolved in ML trees inferred by the quartet puzzling algorithm. A rooting of the RNA polymerase-subunit tree at the mycoplasma level seen in both the MP trees and the ML trees reconstructed with suboptimal amino acid substitution models is not supported by the EF-based phylogenies which robustly affiliate mycoplasmas with low-G+C gram-positives and, most probably, reflects a ``long branch attraction' artifact. Received: 22 September 1999 / Accepted: 11 January 2000     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

REML estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.     

Estimating the number of channels in patch recordings

The estimation of the number of channels in a patch was assumed to be equivalent to the estimation of the binomial parameter n. Seven estimators were evaluated, using data sets simulated for a range of parameters appropriate for single channel recording experiments. No single estimator was best for all parameters; a combination of estimators is a possible option to avoid the biases of individual estimators. All estimators were highly accurate in estimating n in the case that n = 1. For n ≤ 4 the simplest estimator, the maximum number of simultaneously open channels, was the best, For larger values of n the best estimators were Bayesian.     

Mapping of two genes for resistance to clubroot (Plasmodiophora brassicae) in a population of doubled haploid lines of Brassica oleracea by means of RFLP and AFLP markers

A genetic map covering 615 cM in 12 linkage groups was assembled based on 92 RFLP and AFLP markers segregating in a population of 107 doubled haploid lines (DH lines) of Brassica oleracea. The DH-line population was obtained through microspore culture from the of two homozygous parents: DH-line Bi derived from the cabbage landrace Bindsachsener, and DH-line Gr from broccoli cv ‘Greenia’. Sixty-five percent of the loci, and in some cases complete linkage groups, displayed distorted segregation ratios, a frequency much higher than that observed in populations of the same species. DH-line Bi was resistant to clubroot, which is caused by a Dutch field isolate of Plasmodiophora brassicae. Resistance in the DH-line population was determined in two ways: by assigning symptom grades to each plant, and by measuring the fresh weights of the healthy and affected parts of the root system of each plant. Using a multiple QTL mapping approach to analyze the fresh weight data, we found two loci for clubroot resistance; these were designated pb-3 and pb-4. The additive effects of these loci were responsible for 68% of the difference between the parents and for 60% of the genetic variance among DH-line means. Also, indications for the presence of two additional, minor QTLs were found. Analysis of symptom grades revealed the two QTLs pb-3 and pb-4, as well as one of the two minor QTLs indicated by analysis of the fresh weight data. Received: 29 April 1996 / Accepted: 10 May 1996     

Branch Length Heterogeneity Leads to Nonindependent Branch Length Estimates and Can Decrease the Efficiency of Methods of Phylogenetic Inference

Branch length estimates play a central role in maximum-likelihood (ML) and minimum-evolution (ME) methods of phylogenetic inference. For various reasons, branch length estimates are not statistically independent under ML or ME. We studied the response of correlations among branch length estimates to the degree of among-branch length heterogeneity (BLH) in the model (true) tree. The frequency and magnitude of (especially negative) correlations among branch length estimates were both shown to increase as BLH increases under simulation and analytically. For ML, we used the correct model (Jukes–Cantor). For ME, we employed ordinary least-squares (OLS) branch lengths estimated under both simple p-distances and Jukes–Cantor distances, analyzed with and without an among-site rate heterogeneity parameter. The efficiency of ME and ML was also shown to decrease in response to increased BLH. We note that the shape of the true tree will in part determine BLH and represents a critical factor in the probability of recovering the correct topology. An important finding suggests that researchers cannot expect that different branches that were in fact the same length will have the same probability of being accurately reconstructed when BLH exists in the overall tree. We conclude that methods designed to minimize the interdependencies of branch length estimates (BLEs) may (1) reduce both the variance and the covariance associated with the estimates and (2) increase the efficiency of model-based optimality criteria. We speculate on possible ways to reduce the nonindependence of BLEs under OLS and ML. Received: 9 March 1999 / Accepted: 4 May 1999     

Genetic control of macro-and micro-environmental sensitivities in Populus

Understanding the genetic mechanisms for the phenotypic plasticity and developmental instability of a quantitative trait has important implications for breeding and evolution. Two clonally replicated plantations of two 3-generation inbred pedigrees derived from the highly divergent species Populus trichocarpa and P. deltoides were used to examine the genetic control of macro- and micro-environmental sensitivities and their genetic relationships with the trait mean across two contrasting environments. For all stem-growth traits studied, the trait mean had a higher broad-sense heritability (H²) level than macroenvironmental sensitivity, both with much higher values than microenvironmental sensitivity. Genetic correlation analyses indicated that the trait mean was more or less independent of macro- or micro-environmental sensitivity in stem height. Thus, for this trait, the genetic difference in response to the two environments might be mainly due to epistasis between some regulatory loci for plasticity and loci for trait mean. However, for basal area and volume index, pleiotropic loci might be more important for their genetic differences between the two environments. No evidence was found to support Lerner’s (1954) homeostasis theory in which macro- or micro-environmental sensitivity is the inverse function of heterozygosity. Received: 8 March 1996 / Accepted: 31 May 1996     

Making decisions about the optimal selection environment using Gibbs sampling

The use of Gibbs sampling in making decisions about the optimal selection environment was demonstrated. Marginal posterior distributions of the efficiency of selection across sites were obtained using the Gibbs sampler, a Bayesian method, from which the probability that the efficiency of selection lay between specified values and the variance of the distribution were computed, providing a lot of information on which to make decisions regarding the location of genetic tests. The heritability, genetic correlations and efficiencies of selection estimated using REML and Gibbs sampling were similar. However, the latter approach showed that the point estimates of the efficiencies of selection were subject to substantial error. The decision regarding selection at maturity was consistent with that obtained using point estimates from REML, but Gibbs sampling allowed the efficiencies of selection to be interpreted with more confidence. The decision regarding early selection differed from that based on REML point estimates. Generally, the decisions to make early selections at site B for planting at both site B and A, and to make selections at maturity at each individual site, were robust to different priors in the Gibbs sampling. Received: 19 June 2000 / Accepted: 18 October 2000     

Mixed Moment Estimation in Hermite Distribution and its Comparison with Other Estimation Procedures from Sample Size Considerations

In this paper (1) expressions (correct to n^?2 terms) for biases, variances, and covariances of the estimators a and b of Hermite distribution with probability generating function Exp[a(t–1) + b(t–1)] are obtained for two mixed moment estimates; (2) for the biases and variance-covariances, approximate regions of the parameter space (a>0, b>0) have been outlined where a sample of size 100 can be considered as “safe” in the sense that contribution of second order terms in them is 5% of that from the first order term; (3) comparison of the biases and variance-covariances of these two sets of estimators are made with those for the moment estimators, maximum likelihood estimates and the even point estimators for a sample of size 100 using the terms up to order n^?2; (4) the comparisons based on n^?2 terms in (3) have not only provided information on the estimation procedures included in the Hermite distribution, but also demonstrated the importance of higher order terms in the sampling properties of the various alternative techniques for the Hermite distribution.     

Protein Tyrosine Kinase cDNAs from Amphioxus, Hagfish, and Lamprey: Isoform Duplications Around the Divergence of Cyclostomes and Gnathostomes

Animals evolved a variety of gene families involved in cell–cell communication and developmental control by gene duplication and domain shuffling. Each family is made up of several subtypes or subfamilies with distinct structures and functions, which diverged by gene duplications and domain shufflings before the divergence of parazoans and eumetazoans. Since the separation from protostomes, vertebrates expanded the multiplicity of members (isoforms) in the same subfamily by further gene duplications in their early evolution before the fish–tetrapod split. To know the dates of isoform duplications more closely, we have conducted isolation and sequencing cDNAs encoding the fibroblast growth factor receptor, Eph, src, and platelet-derived growth factor receptor subtypes belonging to the protein tyrosine kinase family from Branchiostoma belcheri, an amphioxus, Eptatretus burgeri, a hagfish, and Lampetra reissneri, a lamprey. From a phylogenetic tree of each subfamily inferred from a maximum likelihood (ML) method, together with a bootstrap analysis based on the ML method, we have shown that the isoform duplications frequently occurred in the early evolution of vertebrates around or just before the divergence of cyclostomes and gnathostomes by gene duplications and possibly chromosomal duplications. Received: 28 April 1998 / Accepted: 30 June 1999     

Identifying different types of de-differentiated microspores from indica-japonica F1 hybrids with subspecies-differentiating RFLP probes in rice

The indica, japonica and intermediary types of de-differentiated microspores from indica-japonica hybrids were identified with 11 subspecies-differentiating RELP probes in rice (Oryza sativa L.). The results showed that the distribution of indica, japonica and intermediary types of de-differentiated microspores could be easily detected in a simple and quick way using the RFLP method. Moreover, the microspores from the same hybrid but inoculated onto different media, or microspores from different hybrids when inoculated onto the same medium, often displayed distinctive distribution curves of de-differentiated microspores types, indicating that the media employed in this experiment had high selectivity for the de-differentiation of certain types of microspores. The application of the RELP method to de-differentiated microspore identification is of great theoretical and practical significance in rice doubled-haploid breeding. Received: 27 February 1996 / Accepted: 14 June 1996     

Seed-storage-protein loci in RFLP maps of diploid, tetraploid, and hexaploid wheat

 Linkages between high- and low-molecular-weight (M_r) glutenin, gliadin and triticin loci in diploid, tetraploid and hexaploid wheats were studied by hybridization of restriction fragments with DNA clones and by SDS-PAGE. In tetraploid and hexaploid wheat, DNA fragments hybridizing with a low-M_r glutenin clone were mapped at the XGlu-3 locus in the distal region of the maps of chromosome arms 1AS, 1BS, and 1DS. A second locus, designated XGlu-B2, was detected in the middle of the map of chromosome arm 1BS completely linked to the XGli-B3 gliadin locus. The restriction fragments mapped at this locus were shown to co-segregate with B subunits of low-M_r glutenins in SDS-PAGE in tetraploid wheat, indicating that XGlu-B2 is an active low-M_r glutenin locus. A new locus hybridizing with the low-M_r clone was mapped on the long arm of chromosome 7A^m in diploid wheat. No glutenin protein was found to co-segregate with this new locus. Triticin loci were mapped on chromosome arms 1AS, 1BS, and 1DS. A failure to detect triticin proteins co-segregating with DNA fragments mapped at XTri-B1 locus suggests that this locus is not active. No evidence was found for the existence of Gli-A4, and it is concluded that this locus is probably synonymous with Gli-A3. Recombination was observed within the multigene gliadin family mapped at XGli-A11 (1.2 cM).1 Although these closely linked loci may correspond to the previously named Gli-A1 and Gli-A5 loci, they were temporarily designated XGli-A1.1 and XGli-A1.2 until orthology with Gli-A1 and Gli-A5 is established. Received: 25 March 1997 / Accepted: 23 June 1997     

Comparison of REML and Gibbs sampling estimates of multi-trait genetic parameters in Scots pine

Multi-trait (co)variance estimation is an important topic in plant and animal breeding. In this study we compare estimates obtained with restricted maximum likelihood (REML) and Bayesian Gibbs sampling of simulated data and of three traits (diameter, height and branch angle) from a 26-year-old partial diallel progeny test of Scots pine (Pinus sylvestris L.). Based on the results from the simulated data we can conclude that the REML estimates are accurate but the mode of posterior distributions from the Gibbs sampling can be overestimated depending on the level of the heritability. The mean and median of the posteriors were considerably higher than the expected values of the heritabilities. The confidence intervals calculated with the delta method were biased downwardly. The highest probablity density (HPD) interval provides a better interval estimate, but could be slightly biased at the lower level. Similar differences between REML and Gibbs sampling estimates were found for the Scots pine data. We conclude that further simulation studies are needed in order to evaluate the effect of different priors on (co)variance components in the genetic individual model.