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Plants can express different phenotypic responses following polyploidization, but ploidy-dependent phenotypic variation has so far not been assigned to specific genetic factors. To map such effects, segregating populations at different ploidy levels are required. The availability of an efficient haploid-inducer line in Arabidopsis thaliana allows for the rapid development of large populations of segregating haploid offspring. Because Arabidopsis haploids can be self-fertilised to give rise to homozygous doubled haploids, the same genotypes can be phenotyped at both the haploid and diploid ploidy level. Here, we compared the phenotypes of recombinant haploid and diploid offspring derived from a cross between two late flowering accessions to map genotype x ploidy (GxP) interactions. Ploidy-specific quantitative trait loci (QTLs) were detected at both ploidy levels. This implies that mapping power will increase when phenotypic measurements of monoploids are included in QTL analyses. A multi-trait analysis further revealed pleiotropic effects for a number of the ploidy specific QTLs as well as opposite effects at different ploidy levels for general QTLs. Taken together, we provide evidence of genetic variation between different Arabidopsis accessions being causal for dissimilarities in phenotypic responses to altered ploidy levels, revealing a GxP effect. Additionally, by investigating a population derived from late flowering accessions we revealed a major vernalisation specific QTL for variation in flowering time, countering the historical bias of research in early flowering accessions.

Terminal drought is one of the major constraints to crop production in chickpea (Cicer arietinum L.). In order to map drought tolerance related traits at high resolution, we sequenced multi-parent advanced generation intercross (MAGIC) population using whole genome resequencing approach and phenotyped it under drought stress environments for two consecutive years (2013–14 and 2014–15). A total of 52.02 billion clean reads containing 4.67 TB clean data were generated on the 1136 MAGIC lines and eight parental lines. Alignment of clean data on to the reference genome enabled identification of a total, 932,172 of SNPs, 35,973 insertions, and 35,726 deletions among the parental lines. A high-density genetic map was constructed using 57,180 SNPs spanning a map distance of 1606.69 cM. Using compressed mixed linear model, genome-wide association study (GWAS) enabled us to identify 737 markers significantly associated with days to 50% flowering, days to maturity, plant height, 100 seed weight, biomass, and harvest index. In addition to the GWAS approach, an identity-by-descent (IBD)-based mixed model approach was used to map quantitative trait loci (QTLs). The IBD-based mixed model approach detected major QTLs that were comparable to those from the GWAS analysis as well as some exclusive QTLs with smaller effects. The candidate genes like FRIGIDA and CaTIFY4b can be used for enhancing drought tolerance in chickpea. The genomic resources, genetic map, marker-trait associations, and QTLs identified in the study are valuable resources for the chickpea community for developing climate resilient chickpeas.

This work is motivated by the need to accurately estimate genetic effects over time when analysing data from high-throughput phenotyping (HTP) experiments. The HTP data we deal with here are characterised by phenotypic traits measured multiple times in the presence of spatial and temporal noise and a hierarchical organisation at three levels (populations, genotypes within populations, and plants within genotypes). We propose a feasible one-stage spatio-temporal P-spline-based hierarchical approach to model the evolution of the genetic signal over time on a given phenotype while accounting for spatio-temporal noise. We provide the user with appealing tools that take advantage of the sparse model matrices structure to reduce computational complexity. We illustrate the performance of our method using spatio-temporal simulated data and data from the PhenoArch greenhouse platform at INRAE Montpellier. In the plant breeding context, we show that information extracted for genomic selection purposes from our fitted genotypic curves is similar to those obtained using a comparable two-stage P-spline-based approach.

Large agricultural field trials may display irregular spatial trends that cannot be fully captured by a purely randomization-based analysis. For this reason, paralleling the development of analysis-of-variance procedures for randomized field trials, there is a long history of spatial modeling for field trials, starting with the early work of Papadakis on nearest neighbor analysis, which can be cast in terms of first or second differences among neighboring plot values. This kind of spatial modeling is amenable to a natural extension using splines, as has been demonstrated in recent publications in the field. Here, we consider the P-spline framework, focusing on model options that are easy to implement in linear mixed model packages. Two examples serve to illustrate and evaluate the methods. A key conclusion is that first differences are rather competitive with second differences. A further key observation is that second differences require special attention regarding the representation of the null space of the smooth terms for spatial interaction, and that an unstructured variance–covariance structure is required to ensure invariance to translation and rotation of eigenvectors associated with that null space. We develop a strategy that permits fitting this model with ease, but the approach is more demanding than that needed for fitting models using first differences. Hence, even though in other areas, second differences are very commonly used in the application of P-splines, our conclusion is that with field trials, first differences have advantages for routine use.

A novel case of developing a portable spectral imaging device for kiwifruit analysis is presented. Furthermore, a new complementary spectral image processing strategy combining deep learning and advanced chemometric is proposed for processing the spectral images. The deep learning was used for detection and localisation of harvested fruit in the spectral image while the chemometric modelling was used to predict multiple fruit quality related properties i.e., dry matter and soluble solids content. The developed models were independently validated on fruit harvested from a different orchard as well as on a different variety. The one touch spectral imaging presented in this paper can allow widespread usage of spectral imaging for fresh fruit analysis, particularly benefitting non-experts in spectral imaging and chemometrics to routinely use the spectral imaging for fresh fruit analysis.

Temperature response of stem elongation in wheat grown under field conditions indicated that temperature response is highly heritable and linked to the flowering pathway., In wheat, temperature affects the timing and intensity of stem elongation. Genetic variation for this process is therefore important for adaptation. This study investigates the genetic response to temperature fluctuations during stem elongation and its relationship to phenology and height. Canopy height of 315 wheat genotypes (GABI wheat panel) was scanned twice weekly in the field phenotyping platform (FIP) of ETH Zurich using a LIDAR. Temperature response was modelled using linear regressions between stem elongation and mean temperature in each measurement interval. This led to a temperature-responsive (slope) and a temperature-irresponsive (intercept) component. The temperature response was highly heritable (H2=0.81) and positively related to a later start and end of stem elongation as well as final height. Genome-wide association mapping revealed three temperature-responsive and four temperature-irresponsive quantitative trait loci (QTLs). Furthermore, putative candidate genes for temperature-responsive QTLs were frequently related to the flowering pathway in Arabidopsis thaliana, whereas temperature-irresponsive QTLs corresponded to growth and reduced height genes. In combination with Rht and Ppd alleles, these loci, together with the loci for the timing of stem elongation, accounted for 71% of the variability in height. This demonstrates how high-throughput field phenotyping combined with environmental covariates can contribute to a smarter selection of climate-resilient crops.

Key messageWe evaluate self-organizing maps (SOM) to identify adaptation zones and visualize multi-environment genotypic responses. We apply SOM to multiple traits and crop growth model output of large-scale European sunflower data.AbstractGenotype-by-environment interactions (G × E) complicate the selection of well-adapted varieties. A possible solution is to group trial locations into adaptation zones with G × E occurring mainly between zones. By selecting for good performance inside those zones, response to selection is increased. In this paper, we present a two-step procedure to identify adaptation zones that starts from a self-organizing map (SOM). In the SOM, trials across locations and years are assigned to groups, called units, that are organized on a two-dimensional grid. Units that are further apart contain more distinct trials. In an iterative process of reweighting trial contributions to units, the grid configuration is learnt simultaneously with the trial assignment to units. An aggregation of the units in the SOM by hierarchical clustering then produces environment types, i.e. trials with similar growing conditions. Adaptation zones can subsequently be identified by grouping trial locations with similar distributions of environment types across years. For the construction of SOMs, multiple data types can be combined. We compared environment types and adaptation zones obtained for European sunflower from quantitative traits like yield, oil content, phenology and disease scores with those obtained from environmental indices calculated with the crop growth model Sunflo. We also show how results are affected by input data organization and user-defined weights for genotypes and traits. Adaptation zones for European sunflower as identified by our SOM-based strategy captured substantial genotype-by-location interaction and pointed to trials in Spain, Turkey and South Bulgaria as inducing different genotypic responses.

High throughput phenotyping (HTP) platforms and devices are increasingly used for the characterization of growth and developmental processes for large sets of plant genotypes. Such HTP data require challenging statistical analyses in which longitudinal genetic signals need to be estimated against a background of spatio-temporal noise processes. We propose a two-stage approach for the analysis of such longitudinal HTP data. In a first stage, we correct for design features and spatial trends per time point. In a second stage, we focus on the longitudinal modelling of the spatially corrected data, thereby taking advantage of shared longitudinal features between genotypes and plants within genotypes. We propose a flexible hierarchical three-level P-spline growth curve model, with plants/plots nested in genotypes, and genotypes nested in populations. For selection of genotypes in a plant breeding context, we show how to extract new phenotypes, like growth rates, from the estimated genotypic growth curves and their first-order derivatives. We illustrate our approach on HTP data from the PhenoArch greenhouse platform at INRAE Montpellier and the outdoor Field Phenotyping platform at ETH Zürich.

Construction of genetic linkage maps has become a routine step for mapping quantitative trait loci (QTL), particularly in animal and plant breeding populations. Many multiparental populations have recently been produced to increase genetic diversity and QTL mapping resolution. However, few software packages are available for map construction in these populations. In this paper, we build a general framework for the construction of genetic linkage maps from genotypic data in diploid populations, including bi- and multiparental populations, cross-pollinated (CP) populations, and breeding pedigrees. The framework is implemented as an automatic pipeline called magicMap, where the maximum multilocus likelihood approach utilizes genotypic information efficiently. We evaluate magicMap by extensive simulations and eight real datasets: one biparental, one CP, four multiparent advanced generation intercross (MAGIC), and two nested association mapping (NAM) populations, the number of markers ranging from a few hundred to tens of thousands. Not only is magicMap the only software capable of accommodating all of these designs, it is more accurate and robust to missing genotypes and genotyping errors than commonly used packages.

Yield is a function of environmental quality and the sensitivity with which genotypes react to that. Environmental quality is characterized by meteorological data, soil and agronomic management, whereas genotypic sensitivity is embodied by combinations of physiological traits that determine the crop capture and partitioning of environmental resources over time. This paper illustrates how environmental quality and genotype responses can be studied by a combination of crop simulation and statistical modelling. We characterized the genotype by environment interaction for grain yield of a wheat population segregating for flowering time by simulating it using the the Agricultural Production Systems sIMulator (APSIM) cropping systems model. For sites in the NE Australian wheat-belt, we used meteorological information as integrated by APSIM to classify years according to water, heat and frost stress. Results highlight that the frequency of years with more severe water and temperature stress has largely increased in recent years. Consequently, it is likely that future varieties will need to cope with more stressful conditions than in the past, making it important to select for flowering habits contributing to temperature and water-stress adaptation. Conditional on year types, we fitted yield response surfaces as functions of genotype, latitude and longitude to virtual multi-environment trials. Response surfaces were fitted by two-dimensional P-splines in a mixed-model framework to predict yield at high spatial resolution. Predicted yields demonstrated how relative genotype performance changed with location and year type and how genotype by environment interactions can be dissected. Predicted response surfaces for yield can be used for performance recommendations, quantification of yield stability and environmental characterization.

Large agricultural field trials may display irregular spatial trends that cannot be fully captured by a purely randomization-based analysis. For this reason, paralleling the development of analysis-of-variance procedures for randomized field trials, there is a long history of spatial modelling for field trials, starting with the early work of Papadakis on nearest neighbour analysis, which can be cast in terms of first or second differences among neighbouring plot values. This kind of spatial modelling is amenable to a natural extension using P-splines, as has been demonstrated in recent publications in the field. Here, we consider the P-spline framework, focussing on model options that are easy to implement in linear mixed model packages. Two examples serve to illustrate and evaluate the methods. A key conclusion is that first differences are rather competitive with second differences. A further key observation is that second differences require special attention regarding the representation of the null space of the smooth terms for spatial interaction, and that an unstructured variance-covariance structure is required to ensure invariance to translation and rotation of eigenvectors associated with that null space. We develop a strategy that permits fitting this model with ease, but the approach is more demanding than that needed for fitting models using first differences. Hence, even though in other areas second differences are very commonly used in the application of P-splines, our main conclusion is that with field trials first differences have advantages for routine use.

The aim of this study was to compare a laboratory-based and a pocket-sized near-infrared (NIR) spectrophotometer (SCiO) to predict mango fruit firmness. Three batches of mango fruit were measured to explore batch specific and global models. To gain insight to useful wavelengths important for predicting mango firmness variables, the bootstrapping soft shrinkage (BOSS) variable selection routine was used. Modelling was performed with partial least-squares regression (PLSR). The reference firmness measurements were performed with AWETA acoustic firmness analyser. The SCiO and the laboratory-based instrument showed similar performances predicting the reference firmness in terms of prediction coefficient of determination (R2). However, the root mean squared error of prediction (RMSEP) was slightly lower for the laboratory-based instrument compared to the SCiO, likely because a broader spectral region was used. The performance of the batch specific models was improved by up to 8 % in R2 with a 13 % reduction in RMSEP when BOSS was applied. For both laboratory based and SCiO, the global models based on combined data from the three batches, showed good performance (R2 0.74-0.93, RMSEP 4.8 – 8.2 Hz2g2/3 depending on the batch) to predict the firmness. Due to comparable performance of the SCiO compared to the laboratory-based spectrophotometer, the pocket-sized SCiO NIR sensor has the potential to become a low-cost, easy to use non-destructive tool to measure firmness in mango fruit.

Key message The identity-by-descent (IBD)-based mixed model approach introduced in this study can detect quantitative trait loci (QTLs) referring to the parental origin and simultaneously account for multilevel relatedness of individuals within and across families. This unified approach is proved to be a powerful approach for all kinds of multiparental population (MPP) designs. Abstract Multiparental populations (MPPs) have become popular for quantitative trait loci (QTL) detection. Tools for QTL mapping in MPPs are mostly developed for specific MPPs and do not generalize well to other MPPs. We present an IBD-based mixed model approach for QTL mapping in all kinds of MPP designs, e.g., diallel, Nested Association Mapping (NAM), and Multiparental Advanced Generation Intercross (MAGIC) designs. The first step is to compute identity-by-descent (IBD) probabilities using a general Hidden Markov model framework, called reconstructing ancestry blocks bit by bit (RABBIT). Next, functions of IBD information are used as design matrices, or genetic predictors, in a mixed model approach to estimate variance components for multiallelic genetic effects associated with parents. Family-specific residual genetic effects are added, and a polygenic effect is structured by kinship relations between individuals. Case studies of simulated diallel, NAM, and MAGIC designs proved that the advanced IBD-based multi-QTL mixed model approach incorporating both kinship relations and family-specific residual variances (IBD.MQMkin_F) is robust across a variety of MPP designs and allele segregation patterns in comparison to a widely used benchmark association mapping method, and in most cases, outperformed or behaved at least as well as other tools developed for specific MPP designs in terms of mapping power and resolution. Successful analyses of real data cases confirmed the wide applicability of our IBD-based mixed model methodology.

International audience; Close-range spectral imaging (SI) of agricultural plants is widely performed for digital plant phenotyping. A key task in digital plant phenotyping is the non-destructive and rapid identification of drought stress in plants so as to allow plant breeders to select potential genotypes for breeding drought-resistant plant varieties. Visible and near-infrared SI is a key sensing technique that allows the capture of physicochemical changes occurring in the plant under drought stress. The main challenges are in processing the massive spectral images to extract information relevant for plant breeders to support genotype selection. Hence, this study presents a generic data processing workflow for analysing SI data generated in real-world digital phenotyping experiments to extract meaningful information for decision making by plant breeders. The workflow is a combination of chemometric approaches and deep learning. The usefulness of the proposed workflow is demonstrated on a real-life experiment related to drought stress detection and quantification in Arabidopsis thaliana plants grown in a semi-controlled environment. The results show that the proposed approach is able to detect the presence of drought just 3 days after its induction compared to the well-watered plants. Furthermore, the unsupervised clustering approach provides detailed time-series images where the drought-related changes in plants can be followed visually along the time course. The developed approach facilitates digital phenotyping and can thus accelerate breeding of drought-tolerant plant varieties.

Many different types of multiparental populations have recently been produced to increase genetic diversity and resolution in QTL mapping. Low-coverage, genotyping-by-sequencing (GBS) technology has become a cost-effective tool in these populations, despite large amounts of missing data in offspring and founders. In this work, we present a general statistical framework for genotype imputation in such experimental crosses from low-coverage GBS data. Generalizing a previously developed hidden Markov model for calculating ancestral origins of offspring DNA, we present an imputation algorithm that does not require parental data and that is applicable to bi- and multiparental populations. Our imputation algorithm allows heterozygosity of parents and offspring as well as error correction in observed genotypes. Further, our approach can combine imputation and genotype calling from sequencing reads, and it also applies to called genotypes from SNP array data. We evaluate our imputation algorithm by simulated and real data sets in four different types of populations: the F2, the advanced intercross recombinant inbred lines, the multiparent advanced generation intercross, and the cross-pollinated population. Because our approach uses marker data and population design information efficiently, the comparisons with previous approaches show that our imputation is accurate at even very low (

Nearest-neighbour methods based on first differences are an approach to spatial analysis of field trials with a long history, going back to the early work by Papadakis first published in 1937. These methods are closely related to a geostatistical model that assumes spatial covariance to be a linear function of distance. Recently, P-splines have been proposed as a flexible alternative to spatial analysis of field trials. On the surface, P-splines may appear like a completely new type of method, but closer scrutiny reveals intimate ties with earlier proposals based on first differences and the linear variance model. This paper studies these relations in detail, first focussing on one-dimensional spatial models and then extending to the two-dimensional case. Two yield trial datasets serve to illustrate the methods and their equivalence relations. Parsimonious linear variance and random walk models are suggested as a good point of departure for exploring possible improvements of model fit via the flexible P-spline framework.

Canopy temperature (CT) has been related to water-use and yield formation in crops. However, constantly (e.g., sun illumination angle, ambient temperature) as well as rapidly (e.g., clouds) changing environmental conditions make it difficult to compare measurements taken even at short time intervals. This poses a great challenge for high-throughput field phenotyping (HTFP). The aim of this study was to i) set up a workflow for unmanned aerial vehicles (UAV) based HTFP of CT, ii) investigate different data processing procedures to combine information from multiple images into orthomosaics, iii) investigate the repeatability of the resulting CT by means of heritability, and iv) investigate the optimal timing for thermography measurements. Additionally, the approach was v) compared with other methods for HTFP of CT. The study was carried out in a winter wheat field trial with 354 genotypes planted in two replications in a temperate climate, where a UAV captured CT in a time series of 24 flights during 6 weeks of the grain-filling phase. Custom-made thermal ground control points enabled accurate georeferencing of the data. The generated thermal orthomosaics had a high spatial accuracy (mean ground sampling distance of 5.03 cm/pixel) and position accuracy [mean root-mean-square deviation (RMSE) = 4.79 cm] over all time points. An analysis on the impact of the measurement geometry revealed a gradient of apparent CT in parallel to the principle plane of the sun and a hotspot around nadir. Averaging information from all available images (and all measurement geometries) for an area of interest provided the best results by means of heritability. Correcting for spatial in-field heterogeneity as well as slight environmental changes during the measurements were performed with the R package SpATS. CT heritability ranged from 0.36 to 0.74. Highest heritability values were found in the early afternoon. Since senescence was found to influence the results, it is recommended to measure CT in wheat after flowering and before the onset of senescence. Overall, low-altitude and high-resolution remote sensing proved suitable to assess the CT of crop genotypes in a large number of small field plots as is required in crop breeding and variety testing experiments.

Key message Awn length was mapped using a multiparent population derived from cv. Morex and four wild accessions. One QTL was fine mapped and candidate genes were identified in NILs by RNA-seq. Abstract Barley awns are photosynthetically active and contribute to grain yield. Awn length is variable among both wild and cultivated barley genotypes and many mutants with alterations in awn length have been identified. Here, we used a multiparent mapping population derived from cv. Morex and four genetically diverse wild barley lines to detect quantitative trait loci (QTLs) for awn length. Twelve QTLs, distributed over the barley genome, were identified with the most significant one located on chromosome arm 7HL (QTL AL7.1). The effect of AL7.1 was confirmed using near isogenic lines (NILs) and fine-mapped in two independent heterogeneous inbred families to a

Genotype by environment interaction (G×E) for the target trait, e.g. yield, is an emerging property of agricultural systems and results from the interplay between a hierarchy of secondary traits involving the capture and allocation of environmental resources during the growing season. This hierarchy of secondary traits ranges from basic traits that correspond to response mechanisms/sensitivities, to intermediate traits that integrate a larger number of processes over time and therefore show a larger amount of G×E. Traits underlying yield differ in their contribution to adaptation across environmental conditions and have different levels of G×E. Here, we provide a framework to study the performance of genotype to phenotype (G2P) modeling approaches. We generate and analyze response surfaces, or adaptation landscapes, for yield and yield related traits, emphasizing the organization of the traits in a hierarchy and their development and interactions over time. We use the crop growth model APSIM-wheat with genotype-dependent parameters as a tool to simulate non-linear trait responses over time with complex trait dependencies and apply it to wheat crops in Australia. For biological realism, APSIM parameters were given a genetic basis of 300 QTLs sampled from a gamma distribution whose shape and rate parameters were estimated from real wheat data. In the simulations, the hierarchical organization of the traits and their interactions over time cause G×E for yield even when underlying traits do not show G×E. Insight into how G×E arises during growth and development helps to improve the accuracy of phenotype predictions within and across environments and to optimize trial networks. We produced a tangible simulated adaptation landscape for yield that we first investigated for its biological credibility by statistical models for G×E that incorporate genotypic and environmental covariables. Subsequently, the simulated trait data were used to evaluate statistical genotype-to-phenotype models for multiple traits and environments and to characterize relationships between traits over time and across environments, as a way to identify traits that could be useful to select for specific adaptation. Designed appropriately, these types of simulated landscapes might also serve as a basis to train other, more deep learning methodologies in order to transfer such network models to real-world situations.

New types of phenotyping tools generate large amounts of data on many aspects of plant physiology and morphology with high spatial and temporal resolution. These new phenotyping data are potentially useful to improve understanding and prediction of complex traits, like yield, that are characterized by strong environmental context dependencies, i.e., genotype by environment interactions. For an evaluation of the utility of new phenotyping information, we will look at how this information can be incorporated in different classes of genotype-to-phenotype (G2P) models. G2P models predict phenotypic traits as functions of genotypic and environmental inputs. In the last decade, access to high-density single nucleotide polymorphism markers (SNPs) and sequence information has boosted the development of a class of G2P models called genomic prediction models that predict phenotypes from genome wide marker profiles. The challenge now is to build G2P models that incorporate simultaneously extensive genomic information alongside with new phenotypic information. Beyond the modification of existing G2P models, new G2P paradigms are required. We present candidate G2P models for the integration of genomic and new phenotyping information and illustrate their use in examples. Special attention will be given to the modelling of genotype by environment interactions. The G2P models provide a framework for model based phenotyping and the evaluation of the utility of phenotyping information in the context of breeding programs. ispartof: Plant Science vol:282 pages:23-39 ispartof: location:MEXICO, Int Maize & Wheat Improvement Ctr status: Published online

Chromosome substitution lines (CSLs) are tentatively supreme resources to investigate non-allelic genetic interactions. However, the difficulty of generating such lines in most species largely yielded imperfect CSL panels, prohibiting a systematic dissection of epistasis. Here, we present the development and use of a unique and complete panel of CSLs inArabidopsis thaliana, allowing the full factorial analysis of epistatic interactions. A first comparison of reciprocal single chromosome substitutions revealed a dependency of QTL detection on different genetic backgrounds. The subsequent analysis of the complete panel of CSLs enabled the mapping of the genetic interactors and identified multiple two- and three-way interactions for different traits. Some of the detected epistatic effects were as large as any observed main effect, illustrating the impact of epistasis on quantitative trait variation. We, therefore, have demonstrated the high power of detection and mapping of genome-wide epistasis, confirming the assumed supremacy of comprehensive CSL sets.One sentence summaryDevelopment of a complete panel of chromosome substitution lines enables high power mapping of epistatic interactions inArabidopsis thaliana.

An important aim of the analysis of agricultural field experiments is to obtain good predictions for genotypic performance, by correcting for spatial effects. In practice these corrections turn out to be complicated, since there can be different types of spatial effects; those due to management interventions applied to the field plots and those due to various kinds of erratic spatial trends. This paper explores the use of two-dimensional smooth surfaces to model random spatial variation. We propose the use of anisotropic tensor product P-splines to explicitly model large-scale (global trend) and small-scale (local trend) spatial dependence. On top of this spatial field, effects of genotypes, blocks, replicates, and/or other sources of spatial variation are described by a mixed model in a standard way. Each component in the model is shown to have an effective dimension. They are closely related to variance estimation, and helpful for characterising the importance of model components. An important result of this paper is the formal proof of the relation between several definitions of heritability and the effective dimension associated with the genetic component. The practical value of our approach is illustrated by simulations and analyses of large-scale plant breeding experiments. An \texttt{R}-package, \texttt{SpATS}, is provided

Heritability is a central parameter in quantitative genetics, both from an evolutionary and a breeding perspective. For plant traits heritability is traditionally estimated by comparing within and between genotype variability. This approach estimates broad-sense heritability, and does not account for different genetic relatedness. With the availability of high-density markers there is growing interest in marker based estimates of narrow-sense heritability, using mixed models in which genetic relatedness is estimated from genetic markers. Such estimates have received much attention in human genetics but are rarely reported for plant traits. A major obstacle is that current methodology and software assume a single phenotypic value per genotype, hence requiring genotypic means. An alternative that we propose here, is to use mixed models at individual plant or plot level. Using statistical arguments, simulations and real data we investigate the feasibility of both approaches, and how these affect genomic prediction with G-BLUP and genome-wide association studies. Heritability estimates obtained from genotypic means had very large standard errors and were sometimes biologically unrealistic. Mixed models at individual plant or plot level produced more realistic estimates, and for simulated traits standard errors were up to 13 times smaller. Genomic prediction was also improved by using these mixed models, with up to a 49% increase in accuracy. For GWAS on simulated traits, the use of individual plant data gave almost no increase in power. The new methodology is applicable to any complex trait where multiple replicates of individual genotypes can be scored. This includes important agronomic crops, as well as bacteria and fungi., 6 figures. Contains all the supplementary information (p. 30-54)

Molecular BioSystems 11(11), 3101-3110 (2015). doi:10.1039/C5MB00477B, Published by Royal Society of Chemistry, Cambridge

The potential for using whole genome sequencing (WGS) data in microbiological risk assessment (MRA) has been discussed on several occasions since the beginning of this century. Still, the proposed heuristic approaches have never been applied in a practical framework. This is due to the non-trivial problem of mapping microbial information consisting of thousands of loci onto a probabilistic scale for risks. The paradigm change for MRA involves translation of multidimensional microbial genotypic information to much reduced (integrated) phenotypic information and onwards to a single measure of human risk (i.e. probability of illness). In this paper a first approach in methodology development is described for the application of WGS data in MRA; this is supported by a practical example. That is, combining genetic data (single nucleotide polymorphisms; SNPs) for Shiga toxin-producing Escherichia coli (STEC) O157 with phenotypic data (in vitro adherence to epithelial cells as a proxy for virulence) leads to hazard identification in a Genome Wide Association Study (GWAS). This application revealed practical implications when using SNP data for MRA. These can be summarized by considering the following main issues: optimum sample size for valid inference on population level, correction for population structure, quantification and calibration of results, reproducibility of the analysis, links with epidemiological data, anchoring and integration of results into a systems biology approach for the translation of molecular studies to human health risk. Future developments in genetic data analysis for MRA should aim at resolving the mapping problem of processing genetic sequences to come to a quantitative description of risk. The development of a clustering scheme focusing on biologically relevant information of the microbe involved would be a useful approach in molecular data reduction for risk assessment.

Linkage disequilibrium decay in sugar beet is strongly affected by the breeding history, and varies extensively between and along chromosomes, allowing identification of known and unknown signatures of selection. Genetic diversity and linkage disequilibrium (LD) patterns were investigated in 233 elite sugar beet breeding lines and 91 wild beet accessions, using 454 single nucleotide polymorphisms (SNPs) and 418 SNPs, respectively. Principal coordinate analysis suggested the existence of three groups of germplasm, corresponding to the wild beets, the seed parent and the pollen parent breeding pool. LD was investigated in each of these groups, with and without correction for genetic relatedness. Without correction for genetic relatedness, in the pollen as well as the seed parent pool, LD persisted beyond 50 centiMorgan (cM) on four (2, 3, 4 and 5) and three chromosomes (2, 4 and 6), respectively; after correction for genetic relatedness, LD decayed after

Adjustment for spatial trends in plant breeding field trials is essential for efficient evaluation and selection of genotypes. Current mixed model methods of spatial analysis are based on a multi-step modelling process where global and local trends are fitted after trying several candidate spatial models. This paper reports the application of a novel spatial method that accounts for all types of continuous field variation in a single modelling step by fitting a smooth surface. The method uses two-dimensional P-splines with anisotropic smoothing formulated in the mixed model framework, referred to as SpATS model. We applied this methodology to a series of large and partially replicated sorghum breeding trials. The new model was assessed in comparison with the more elaborate standard spatial models that use autoregressive correlation of residuals. The improvements in precision and the predictions of genotypic values produced by the SpATS model were equivalent to those obtained using the best fitting standard spatial models for each trial. One advantage of the approach with SpATS is that all patterns of spatial trend and genetic effects were modelled simultaneously by fitting a single model. Furthermore, we used a flexible model to adequately adjust for field trends. This strategy reduces potential parameter identification problems and simplifies the model selection process. Therefore, the new method should be considered as an efficient and easy-to-use alternative for routine analyses of plant breeding trials., MTM2014-55966-P Grains Research and Development Corporation (GRDC) of Australia National Institute of Agricultural Technology (INTA) of Argentina, Res. DN 1126/13 Integrated Breeding Program

The common assumption in quantitative trait locus (QTL) linkage mapping studies that parents of multiple connected populations are unrelated is unrealistic for many plant breeding programs. We remove this assumption and propose a Bayesian approach that clusters the alleles of the parents of the current mapping populations from locus-specific identity by descent (IBD) matrices that capture ancestral marker and pedigree information. Moreover, we demonstrate how the parental IBD data can be incorporated into a QTL linkage analysis framework by using two approaches: a Threshold IBD model (TIBD) and a Latent Ancestral Allele Model (LAAM). The TIBD and LAAM models are empirically tested via numerical simulation based on the structure of a commercial maize breeding program. The simulations included a pilot dataset with closely linked QTL on a single linkage group and 100 replicated datasets with five linkage groups harboring four unlinked QTL. The simulation results show that including parental IBD data (similarly for TIBD and LAAM) significantly improves the power and particularly accuracy of QTL mapping, e.g., position, effect size and individuals’ genotype probability without significantly increasing computational demand.

25 Pag., 5 Tabl., 6 Fig. The definitive version is available at: www.springerlink.com, Quantitative trait locus (QTL) detection is commonly performed by analysis of designed segregating populations derived from two inbred parental lines, where absence of selection, mutation and genetic drift is assumed. Even for designed populations, selection cannot always be avoided, with as consequence varying correlation between genotypes instead of uniform correlation. Akin to linkage disequilibrium mapping, ignoring this type of genetic relatedness will increase the rate of false-positives. In this paper, we advocate using mixed models including genetic relatedness, or ‘kinship’ information for QTL detection in populations where selection forces operated. We demonstrate our case with a three-way barley cross, designed to segregate for dwarfing, vernalization and spike morphology genes, in which selection occurred. The population of 161 inbred lines was screened with 1,536 single nucleotide polymorphisms (SNPs), and used for gene and QTL detection. The coefficient of coancestry matrix was estimated based on the SNPs and imposed to structure the distribution of random genotypic effects. The model incorporating kinship, coancestry, information was consistently superior to the one without kinship (according to the Akaike information criterion). We show, for three traits, that ignoring the coancestry information results in an unrealistically high number of marker–trait associations, without providing clear conclusions about QTL locations. We used a number of widely recognized dwarfing and vernalization genes known to segregate in the studied population as landmarks or references to assess the agreement of the mapping results with a priori candidate gene expectations. Additional QTLs to the major genes were detected for all traits as well., We want to thank INIA (MICINN) for partially funding this work through different grants. The Centre UdL-IRTA forms part of the Centre CONSOLIDER on Agrigenomics funded by the Spanish Ministry of Education and Science and acknowledges partial funding from grant AGL2005-07195-C02-02. Genotyping of the RIL population with BOPA1 was funded by the Spanish Ministry of Science and Innovation, project GEN2006-28560-E. The work of M.M. and FvE was partially financed by the Generation Challenge Programme, through the Integrated Breeding Platform projects ‘2.2.1. Develop and deploy statistical and genetic analysis methodology for molecular breeding’ and ‘3.2.4. Design and Analysis’. The research of FvE was partly financed by project BB9 of the Centre for Bio-Systems Genomics (CBSG), which is part of the Netherlands Genomics Initiative/Netherlands Organization for Scientific Research. This work was carried out within the research programme of the Netherlands Consortium for Systems Biology (NCSB), which is part of the Netherlands Genomics Initiative / Netherlands Organization for Scientific Research.

Food chain models of ordinary differential equations (ode's) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil'nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections 'cut' the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models. © 2010 Elsevier Inc.

Genetic linkage and association studies are empowered by proper modeling of relatedness among individuals. Such relatedness can be inferred from marker and/or pedigree information. In this study, the genetic relatedness among n inbred individuals at a particular locus is expressed as an n × n square matrix Q. The elements of Q are identity-by-descent probabilities, that is, probabilities that two individuals share an allele descended from a common ancestor. In this representation the definition of the ancestral alleles and their number remains implicit. For human inspection and further analysis, an explicit representation in terms of the ancestral allele origin and the number of alleles is desirable. To this purpose, we decompose the matrix Q by a latent class model with K classes (latent ancestral alleles). Let P be an n × K matrix with assignment probabilities of n individuals to K classes constrained such that every element is nonnegative and each row sums to 1. The problem then amounts to approximating Q by PPT, while disregarding the diagonal elements. This is not an eigenvalue problem because of the constraints on P. An efficient algorithm for calculating P is provided. We indicate the potential utility of the latent ancestral allele model. For representative locus-specific Q matrices constructed for a set of maize inbreds, the proposed model recovered the known ancestry.

A Bayesian approach to analyze complex traits is presented that can help plant eneticists and breeders in exploiting the marker and phenotypic data on pedigreed populations as available from ongoing breeding programs. The statistical model for the quantitative trait may include non-genetic and genetic components. The latter component can be divided into QTL on known marker linkage groups, major genes and a polygenic component. The full probability model, prior assumptions on model variables are presented and criterion for model selection and posterior inferences are given. Simulated data on a known pedigreed population structure of the EU project HiDRAS was used to illustrate the use of the Bayesian approach to analyze complex traits. It was shown that estimates for QTL parameters were more accurate when non-genetic factors were included in the model and when a polygenic component was included when not all linkage groups were analyzed simultaneously. The Bayesian approach has been implemented into the software package FlexQTL and allows plant breeders explore their pedigreed populations for segregating QTL alleles that are relevant in their breeding program.

Prediction of the phenotypes for a set of genotypes across multiple environments is a fundamental task in any plant breeding program. Genomic prediction (GP) can assist selection decisions by combining incomplete phenotypic information over multiple environments (MEs) with dense sets of markers. We compared a range of ME-GP models differing in the way environment-specific genetic effects were modeled. Information among environments was shared either implicitly via the response variable, or by the introduction of explicit environmental covariables. We discuss the models not only in the light of their accuracy, but also in their ability to predict the different parts of the incomplete genotype × environment interaction (G × E) table: (G t ; E t ), (G u ; E t ), (G t ; E u ), and (G u ; E u ), where G is genotype, E is environment, both tested (t; in one or more instances) and untested (u). Using the ‘Steptoe’ × ‘Morex’ barley (Hordeum vulgare L.) population as an example, we show the advantage of ME-GP models that account for G × E. In addition, for our example data set, we show that for prediction in the most challenging scenario of untested environments (E u ), the use of explicit environmental information is preferable over the simpler approach of predicting from a main effects model. Besides producing the most general ME-GP model, the use of environmental covariables naturally links with ecophysiological and crop-growth models (CGMs) for G × E. We conclude with a list of future research topics in ME-GP, where we see CGMs playing a central role.

We present a general hidden Markov model framework called reconstructing ancestry blocks bit by bit (RABBIT) for reconstructing genome ancestry blocks from single-nucleotide polymorphism (SNP) array data, a required step for quantitative trait locus (QTL) mapping. The framework can be applied to a wide range of mapping populations such as the Arabidopsis multiparent advanced generation intercross (MAGIC), the mouse Collaborative Cross (CC), and the diversity outcross (DO) for both autosomes and X chromosomes if they exist. The model underlying RABBIT accounts for the joint pattern of recombination breakpoints between two homologous chromosomes and missing data and allelic typing errors in the genotype data of both sampled individuals and founders. Studies on simulated data of the MAGIC and the CC and real data of the MAGIC, the DO, and the CC demonstrate that RABBIT is more robust and accurate in reconstructing recombination bin maps than some commonly used methods.

In 2003, Xu obtained remarkably precise estimates of QTL positions despite the many markers simultaneously in his Bayesian model. We extend his model and Gibbs algorithm to ensure a valid posterior distribution and convergence to it, without changing the attractiveness of the method.

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level. © 2001 Elsevier Science Inc.

The use of bifurcation analysis for the study of the long—term dynamic behaviour of food webs is discussed. Food webs are collections of populations with often very complicated interactions. The non—viable nutrients are consumed by populations which are consumed by other populations at higher trophic levels, except for the top—predator species. In the most simple setting the dynamics of each population is described mathematically by an ordinary differential equation (ODE). In this paper we give an overview of local as well as global bifurcations found for a simple food web model. With chaotic behaviour all points in a next—return map appear to lie close to single curve. A cubic map with two critical points is studied and the results obtained are used to clarify global bifurcations of the ODE system. These global bifurcations are homoclinic bifurcations of a saddle limit cycle and a heteroclinic bifurcation from the equilibrium to the limit cycle. The homoclinic bifurcation of the saddle limit cycle is ass...

We study the invasion of a top predator into a food chain in a chemostat. For each trophic level, a bioenergetic model is used in which maintenance and energy reserves are taken into account. Bifurcation analysis is performed on the set of nonlinear ordinary differential equations which describe the dynamic behaviour of the food chain. In this paper, we analyse how the ability of a top predator to invade the food chain depends on the values of two control parameters: the dilution rate and the concentration of the substrate in the input. We investigate invasion by studying the long- term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.

In food chain models the lowest trophic level is often assumed to grow logistically. Anomalous behaviour of the solution of the logistic equation and problems with the introduction of mortality have recently been reported. As predation on the lowest trophic level is a kind of mortality, one expects problems with these food chain models. In this paper we compare two formulations for the lowest trophic level: the logistic growth formulation and the mass balance formulation with resources modelled explicitly. We examine the effects of both models on the dynamic behaviour of a tri-trophic microbial food chain in a chemostat. For this purpose bifurcation diagrams, which give the existence and stability of the equilibria of the nonlinear dynamic system, are used. It turns out that the dynamic behaviours differ in a rather large region of the control parameter space spanned by the dilution rate and the concentration of the resources in the reservoir. We urge that mass balance equations should be used in modelling food chains in chemostats as well as in ecosystems.

The next generation of QTL (quantitative trait loci) mapping populations have been designed with multiple founders, where one to a number of generations of intercrossing are introduced prior to the inbreeding phase to increase accumulated recombinations and thus mapping resolution. Examples of such populations are Collaborative Cross (CC) in mice and Multiparent Advanced Generation Inter-Cross (MAGIC) lines in Arabidopsis. The genomes of the produced inbred lines are fine-grained random mosaics of the founder genomes. In this article, we present a novel framework for modeling ancestral origin processes along two homologous autosomal chromosomes from mapping populations, which is a major component in the reconstruction of the ancestral origins of each line for QTL mapping. We construct a general continuous time Markov model for ancestral origin processes, where the rate matrix is deduced from the expected densities of various types of junctions (recombination breakpoints). The model can be applied to monoecious populations with or without self-fertilizations and to dioecious populations with two separate sexes. The analytic expressions for map expansions and expected junction densities are obtained for mapping populations that have stage-wise constant mating schemes, such as CC and MAGIC. Our studies on the breeding design of MAGIC populations show that the intercross mating schemes do not matter much for large population size and that the overall expected junction density, and thus map resolution, are approximately proportional to the inverse of the number of founders.