Constructing genetic linkage maps under a tetrasomic model

Multiallelic models for QTL mapping in diverse polyploid populations

Abstract

Quantitative trait locus (QTL) analysis allows to identify regions responsible for a trait and to associate alleles with their effect on phenotypes. When using biallelic markers to find these QTL regions, two alleles per QTL are modelled. This assumption might be close to reality in specific biparental crosses but is unrealistic in situations where broader genetic diversity is studied. Diversity panels used in genome-wide association studies or multi-parental populations can easily harbour multiple QTL alleles at each locus, more so in the case of polyploids that carry more than two alleles per individual. In such situations a multiallelic model would be closer to reality, allowing for different genetic effects for each potential allele in the population. To obtain such multiallelic markers we propose the usage of haplotypes, concatenations of nearby SNPs. We developed “mpQTL” an R package that can perform a QTL analysis at any ploidy level under biallelic and multiallelic models, depending on the marker type given. We tested the effect of genetic diversity on the power and accuracy difference between bi-allelic and multiallelic models using a set of simulated multiparental autotetraploid, outbreeding populations. Multiallelic models had higher detection power and were more precise than biallelic, SNP-based models, particularly when genetic diversity was higher. This confirms that moving to multi-allelic QTL models can lead to improved detection and characterization of QTLs.

Key message

QTL detection in populations with more than two functional QTL alleles (which is likely in multiparental and/or polyploid populations) is more powerful when using multiallelic models, rather than biallelic models.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-022-04607-z.

Haplotype reconstruction in connected tetraploid F1 populations

In diploid species, many multiparental populations have been developed to increase genetic diversity and quantitative trait loci (QTL) mapping resolution. In these populations, haplotype reconstruction has been used as a standard practice to increase the power of QTL detection in comparison with the marker-based association analysis. However, such software tools for polyploid species are few and limited to a single biparental F1 population. In this study, a statistical framework for haplotype reconstruction has been developed and implemented in the software PolyOrigin for connected tetraploid F1 populations with shared parents, regardless of the number of parents or mating design. Given a genetic or physical map of markers, PolyOrigin first phases parental genotypes, then refines the input marker map, and finally reconstructs offspring haplotypes. PolyOrigin can utilize single nucleotide polymorphism (SNP) data coming from arrays or from sequence-based genotyping; in the latter case, bi-allelic read counts can be used (and are preferred) as input data to minimize the influence of genotype calling errors at low depth. With extensive simulation we show that PolyOrigin is robust to the errors in the input genotypic data and marker map. It works well for various population designs with ≥30 offspring per parent and for sequences with read depth as low as 10x. PolyOrigin was further evaluated using an autotetraploid potato dataset with a 3 × 3 half-diallel mating design. In conclusion, PolyOrigin opens up exciting new possibilities for haplotype analysis in tetraploid breeding populations.

A tetrasomic inheritance model and likelihood-based method for mapping quantitative trait loci in autotetraploid species

Dissecting the genetic architecture of quantitative traits in autotetraploid species is a methodologically challenging task, but a pivotally important goal for breeding globally important food crops, including potato and blueberry, and ornamental species such as rose. Mapping quantitative trait loci (QTLs) is now a routine practice in diploid species but is far less advanced in autotetraploids, largely due to a lack of analytical methods that account for the complexities of tetrasomic inheritance. We present a novel likelihood-based method for QTL mapping in outbred segregating populations of autotetraploid species. The method accounts properly for sophisticated features of gene segregation and recombination in an autotetraploid meiosis. It may model and analyse molecular marker data with or without allele dosage information, such as that from microarray or sequencing experiments. The method developed outperforms existing bivalent-based methods, which may fail to model and analyse the full spectrum of experimental data, in the statistical power of QTL detection, and accuracy of QTL location, as demonstrated by an intensive simulation study and analysis of data sets collected from a segregating population of potato (Solanum tuberosum). The study enables QTL mapping analysis to be conducted in autotetraploid species under a rigorous tetrasomic inheritance model.

High-Resolution Linkage Map With Allele Dosage Allows the Identification of Regions Governing Complex Traits and Apospory in Guinea Grass (Megathyrsus maximus)

Forage grasses are mainly used in animal feed to fatten cattle and dairy herds, and guinea grass (Megathyrsus maximus) is considered one of the most productive of the tropical forage crops that reproduce by seeds. Due to the recent process of domestication, this species has several genomic complexities, such as autotetraploidy and aposporous apomixis. Consequently, approaches that relate phenotypic and genotypic data are incipient. In this context, we built a linkage map with allele dosage and generated novel information of the genetic architecture of traits that are important for the breeding of M. maximus. From a full-sib progeny, a linkage map containing 858 single nucleotide polymorphism (SNP) markers with allele dosage information expected for an autotetraploid was obtained. The high genetic variability of the progeny allowed us to map 10 quantitative trait loci (QTLs) related to agronomic traits, such as regrowth capacity and total dry matter, and 36 QTLs related to nutritional quality, which were distributed among all homology groups (HGs). Various overlapping regions associated with the quantitative traits suggested QTL hotspots. In addition, we were able to map one locus that controls apospory (apo-locus) in HG II. A total of 55 different gene families involved in cellular metabolism and plant growth were identified from markers adjacent to the QTLs and APOSPORY locus using the Panicum virgatum genome as a reference in comparisons with the genomes of Arabidopsis thaliana and Oryza sativa. Our results provide a better understanding of the genetic basis of reproduction by apomixis and traits important for breeding programs that considerably influence animal productivity as well as the quality of meat and milk.

Linkage Analysis and Haplotype Phasing in Experimental Autopolyploid Populations with High Ploidy Level Using Hidden Markov Models

Modern SNP genotyping technologies allow measurement of the relative abundance of different alleles for a given locus and consequently estimation of their allele dosage, opening a new road for genetic studies in autopolyploids. Despite advances in genetic linkage analysis in autotetraploids, there is a lack of statistical models to perform linkage analysis in organisms with higher ploidy levels. In this paper, we present a statistical method to estimate recombination fractions and infer linkage phases in full-sib populations of autopolyploid species with even ploidy levels for a set of SNP markers using hidden Markov models. Our method uses efficient two-point procedures to reduce the search space for the best linkage phase configuration and reestimate the final parameters by maximizing the likelihood of the Markov chain. To evaluate the method, and demonstrate its properties, we rely on simulations of autotetraploid, autohexaploid and autooctaploid populations and on a real tetraploid potato data set. The results show the reliability of our approach, including situations with complex linkage phase scenarios in hexaploid and octaploid populations.

Linkage Analysis and Haplotype Phasing in Experimental Autopolyploid Populations with High Ploidy Level Using Hidden Markov Models

Modern SNP genotyping technologies allow measurement of the relative abundance of different alleles for a given locus and consequently estimation of their allele dosage, opening a new road for genetic studies in autopolyploids. Despite advances in genetic linkage analysis in autotetraploids, there is a lack of statistical models to perform linkage analysis in organisms with higher ploidy levels. In this paper, we present a statistical method to estimate recombination fractions and infer linkage phases in full-sib populations of autopolyploid species with even ploidy levels for a set of SNP markers using hidden Markov models. Our method uses efficient two-point procedures to reduce the search space for the best linkage phase configuration and reestimate the final parameters by maximizing the likelihood of the Markov chain. To evaluate the method, and demonstrate its properties, we rely on simulations of autotetraploid, autohexaploid and autooctaploid populations and on a real tetraploid potato data set. The results show the reliability of our approach, including situations with complex linkage phase scenarios in hexaploid and octaploid populations.

Quantifying the Power and Precision of QTL Analysis in Autopolyploids Under Bivalent and Multivalent Genetic Models

New genotyping technologies, offering the possibility of high genetic resolution at low cost, have helped fuel a surge in interest in the genetic analysis of polyploid species. Nevertheless, autopolyploid species present extra challenges not encountered in diploids and allopolyploids, such as polysomic inheritance or double reduction. Here we investigate the power and precision of quantitative trait locus (QTL) analysis in outcrossing autopolyploids, comparing the results of a model that assumes random bivalent chromosomal pairing during meiosis to one that also allows for multivalents and double reduction. Through a series of simulation studies we found that marginal gains in QTL detection power are achieved using the double reduction model when multivalent pairing occurs. However, when exploring the effect of variable genotypic information across parental homologs, we found that both QTL detection power and precision require high and uniform genotypic information contents. This effect far outweighed considerations regarding bivalent or multivalent pairing (and double reduction) during meiosis. We propose that autopolyploid QTL studies be accompanied by both marker coverage information and per-homolog genotypic information coefficients (GIC). Application of these methods to an autotetraploid potato (Solanum tuberosum L.) mapping population confirmed our ability to locate and dissect QTL in highly heterozygous outcrossing autotetraploid populations.

Orthogonal contrast based models for quantitative genetic analysis in autotetraploid species

Dissecting the genetic architecture of quantitative traits is a crucial goal for efficient breeding of polyploid plants, including autotetraploid crop species, such as potato and coffee, and ornamentals such as rose. To meet this goal, a quantitative genetic model is needed to link the genetic effects of genes or genotypes at quantitative trait loci (QTL) to the phenotype of quantitative traits. We present a statistically tractable quantitative genetic model for autotetraploids based on orthogonal contrast comparisons in the general linear model. The new methods are suitable for autotetraploid species with any population genetic structure and take full account of the essential features of autotetrasomic inheritance. The statistical properties of the new methods are explored and compared to an alternative method in the literature by simulation studies. We have shown how these methods can be applied for quantitative genetic analysis in autotetraploids by analysing trait phenotype data from an autotetraploid potato segregating population. Using trait segregation analysis, we showed that both highly heritable traits of flowering time and plant height were under the control of major QTL. The orthogonal model directly dissects genetic variance into independent components and gives consistent estimates of genetic effects provided that tetrasomic gene segregation is considered.

AGHmatrix: R Package to Construct Relationship Matrices for Autotetraploid and Diploid Species: A Blueberry Example

Progress in the rate of improvement in autopolyploid species has been limited compared with diploids, mainly because software and methods to apply advanced prediction and selection methodologies in autopolyploids are lacking. The objectives of this research were to (i) develop an R package for autopolyploids to construct the relationship matrix derived from pedigree information that accounts for autopolyploidy and double reduction and (ii) use the package to estimate the level and effect of double reduction in an autotetraploid blueberry breeding population with extensive pedigree information. The package is unique, as it can create A-matrices for different levels of ploidy and double reduction, which can then be used by breeders to fit mixed models in the context of predicting breeding values (BVs). Using the data from this blueberry population, we found for all the traits that tetrasomic inheritance creates a better fit than disomic inheritance. In one of the five traits studied, the level of double reduction was different from zero, decreasing the estimated heritability, but it did not affect the prediction of BVs. We also discovered that different depths of pedigree would have significant implications on the estimation of double reduction using this approach. This freely available R package is available for autopolyploid breeders to estimate the level of double reduction present in their populations and the impact in the estimation of genetic parameters as well as to use advanced methods of prediction and selection.

Integrating haplotype-specific linkage maps in tetraploid species using SNP markers

Linkage mapping can help unravel the complexities of polyploid genomes. Here, we integrate haplotype-specific linkage maps in autotetraploid potato and explore the possibilities for mapping in other polyploid species. High-density linkage mapping in autopolyploid species has become possible in recent years given the increasing number of molecular markers now available through modern genotyping platforms. Such maps along with larger experimental populations are needed before we can obtain sufficient accuracy to make marker-trait association studies useful in practice. Here, we describe a method to create genetic linkage maps for an autotetraploid species with large numbers of markers and apply it to an F1 population of tetraploid potato (Solanum tuberosum L.) of 235 individuals genotyped using a 20K SNP array. SNP intensity values were converted to allele dosages after which we calculated pairwise maximum likelihood estimates of recombination frequencies between all marker segregation types under the assumption of random bivalent pairing. These estimates were used in the clustering of markers into linkage groups and their subsequent ordering into 96 homologue maps. The homologue maps were integrated per chromosome, resulting in a total map length of 1061 cM from 6910 markers covering all 12 potato chromosomes. We examined the questions of marker phasing and binning and propose optimal strategies for both. We also investigated the effect of quadrivalent formation and preferential pairing on recombination frequency estimation and marker phasing, which is of great relevance not only for potato but also for genetic studies in other tetraploid species for which the meiotic pairing behaviour is less well understood.

Genetic mapping with an inbred line-derived F2 population in potato

This is the first report of the production and use of a diploid inbred line-based F2 population for genetic mapping in potato. Potato (Solanum tuberosum L.) is an important global food crop, for which tetrasomic inheritance and self-incompatibility have limited both genetic discovery and breeding gains. We report here on the creation of the first diploid inbred line-derived F2 population in potato, and demonstrate its utility for genetic mapping. To create the population, the doubled monoploid potato DM1-3 was crossed as a female to M6, an S7 inbred line derived from the wild relative S. chacoense, and a single F1 plant was then self-pollinated. A genetic linkage map with 2264 single nucleotide polymorphisms was constructed and used to improve the physical anchoring of superscaffolds in the potato reference genome, which is based on DM1-3. Segregation was observed for skin and flesh color, skin and flesh pigment intensity, tuber shape, anther development, jelly end, and the presence of eye tubers instead of normal sprouts. Using the R/qtl software, we detected 10 genes, 7 of which have been previously mapped and 3 for which this is the first publication. The latter category includes tightly linked genes for the jelly end and eye tuber traits on chromosome 5. The development of recombinant inbred lines from this F2 population by single-seed descent is underway and should facilitate even better resolution of these and other loci.

Probabilistic Multilocus Haplotype Reconstruction in Outcrossing Tetraploids

For both plant (e.g., potato) and animal (e.g., salmon) species, unveiling the genetic architecture of complex traits is key to the genetic improvement of polyploids in agriculture. F1 progenies of a biparental cross are often used for quantitative trait loci (QTL) mapping in outcrossing polyploids, where haplotype reconstruction by identifying the parental origins of marker alleles is necessary. In this paper, we build a novel and integrated statistical framework for multilocus haplotype reconstruction in a full-sib tetraploid family from biallelic marker dosage data collected from single-nucleotide polymorphism (SNP) arrays or next-generation sequencing technology given a genetic linkage map. Compared to diploids, in tetraploids, additional complexity needs to be addressed, including double reduction and possible preferential pairing of chromosomes. We divide haplotype reconstruction into two stages: parental linkage phasing for reconstructing the most probable parental haplotypes and ancestral inference for probabilistically reconstructing the offspring haplotypes conditional on the reconstructed parental haplotypes. The simulation studies and the application to real data from potato show that the parental linkage phasing is robust to, and that the subsequent ancestral inference is accurate for, complex chromosome pairing behaviors during meiosis, various marker segregation types, erroneous genetic maps except for long-range disturbances of marker ordering, various amounts of offspring dosage errors (up to ∼20%), and various fractions of missing data in parents and offspring dosages.

The Double-Reduction Landscape in Tetraploid Potato as Revealed by a High-Density Linkage Map

The creation of genetic linkage maps in polyploid species has been a long-standing problem for which various approaches have been proposed. In the case of autopolyploids, a commonly used simplification is that random bivalents form during meiosis. This leads to relatively straightforward estimation of recombination frequencies using maximum likelihood, from which a genetic map can be derived. However, autopolyploids such as tetraploid potato (Solanum tuberosum L.) may exhibit additional features, such as double reduction, not normally encountered in diploid or allopolyploid species. In this study, we produced a high-density linkage map of tetraploid potato and used it to identify regions of double reduction in a biparental mapping population. The frequency of multivalents required to produce this degree of double reduction was determined through simulation. We also determined the effect that multivalents or preferential pairing between homologous chromosomes has on linkage mapping. Low levels of multivalents or preferential pairing do not adversely affect map construction when highly informative marker types and phases are used. We reveal the double-reduction landscape in tetraploid potato, clearly showing that this phenomenon increases with distance from the centromeres.

Quantitative SNP genotyping of polyploids with MassARRAY and other platforms

Accurate genotyping is essential for building genetic maps and performing genome assembly of polyploid species. Recent high-throughput techniques, such as Illumina GoldenGate™ and Sequenom iPLEX MassARRAY®, have made it possible to accurately estimate the relative abundances of different alleles even when the ploidy of the population is unknown. Here we describe the experimental methods for collecting these relative allele intensities and then demonstrate the practical concerns for inferring genotypes using Bayesian inference via the software package SuperMASSA.

Improving breeding efficiency in potato using molecular and quantitative genetics

Potatoes are highly heterozygous and the conventional breeding of superior germplasm is challenging, but use of a combination of MAS and EBVs can accelerate genetic gain. Cultivated potatoes are highly heterozygous due to their outbreeding nature, and suffer acute inbreeding depression. Modern potato cultivars also exhibit tetrasomic inheritance. Due to this genetic heterogeneity, the large number of target traits and the specific requirements of commercial cultivars, potato breeding is challenging. A conventional breeding strategy applies phenotypic recurrent selection over a number of generations, a process which can take over 10 years. Recently, major advances in genetics and molecular biology have provided breeders with molecular tools to accelerate gains for some traits. Marker-assisted selection (MAS) can be effectively used for the identification of major genes and quantitative trait loci that exhibit large effects. There are also a number of complex traits of interest, such as yield, that are influenced by a large number of genes of individual small effect where MAS will be difficult to deploy. Progeny testing and the use of pedigree in the analysis can provide effective identification of the superior genetic factors that underpin these complex traits. Recently, it has been shown that estimated breeding values (EBVs) can be developed for complex potato traits. Using a combination of MAS and EBVs for simple and complex traits can lead to a significant reduction in the length of the breeding cycle for the identification of superior germplasm.

Improving the analysis of low heritability complex traits for enhanced genetic gain in potato

Best linear unbiased prediction (BLUP), which uses pedigree to estimate breeding values, can result in increased genetic gains for low heritability traits in autotetraploid potato. Conventional potato breeding strategies, based on outcrossing followed by phenotypic recurrent selection over a number of generations, can result in slow but steady improvements of traits with moderate to high heritability. However, faster gains, particularly for low heritability traits, could be made by selection on estimated breeding values (EBVs) calculated using more complete pedigree information in best linear unbiased prediction (BLUP) analysis. One complication in applying BLUP predictions of breeding value to potato breeding programs is the autotetraploid inheritance pattern of this species. Here we have used a large pedigree, dating back to 1908, to estimate heritability for nine key traits for potato breeding, modelling autotetraploid inheritance. We estimate the proportion of double reduction in potatoes from our data, and across traits, to be in the order of 10 %. Estimates of heritability ranged from 0.21 for breeder's visual preference, 0.58 for tuber yield, to 0.83 for plant maturity. Using the accuracies of the EBVs determined by cross generational validation, we model the genetic gain that could be achieved by selection of genotypes for breeding on BLUP EBVs and demonstrate that gains can be greater than in conventional schemes.

Natural pathways to polyploidy in plants and consequences for genome reorganization

The last decade highlighted polyploidy as a rampant evolutionary process that triggers drastic genome reorganization, but much remains to be understood about their causes and consequences in both autopolyploids and allopolyploids. Here, we provide an overview of the current knowledge on the pathways leading to different types of polyploids and patterns of polyploidy-induced genome restructuring and functional changes in plants. Available evidence leads to a tentative 'diverge, merge and diverge' model supporting polyploid speciation and stressing patterns of divergence between diploid progenitors as a suitable predictor of polyploid genome reorganization. The merging of genomes at the origin of a polyploid lineage may indeed reveal different kinds of incompatibilities (chromosomal, genic and transposable elements) that have accumulated in diverging progenitors and reduce the fitness of nascent polyploids. Accordingly, successful polyploids have to overcome these incompatibilities through non-Mendelian mechanisms, fostering polyploid genome reorganization in association with the establishment of new lineages. See also sister article focusing on animals by Collares-Pereira et al., in this themed issue.

The simulation of meiosis in diploid and tetraploid organisms using various genetic models

Background

While the genetics of diploid inheritance are well studied and software for linkage mapping, haplotyping and QTL analysis are available, for tetraploids the available tools are limited. In order to develop such tools it would be helpful if simulated populations based on a variety of models of the tetraploid meiosis would be available.

Results

Here we present PedigreeSim, a software package that simulates meiosis in both diploid and tetraploid species and uses this to simulate pedigrees and cross populations. For tetraploids a variety of models can be used, including both bivalent and quadrivalent formation, varying degrees of preferential pairing of hom(oe)ologous chromosomes, different quadrivalent configurations and more. Simulation of quadrivalent meiosis results as expected in double reduction and recombination between more than two hom(oe)ologous chromosomes. The results are shown to match theoretical predictions.

Conclusions

This is the first simulation software that implements all features of meiosis in tetraploids. It allows to generate data for tetraploid and diploid populations, and to investigate different models of tetraploid meiosis. The software and manual are available from http://www.plantbreeding.nl/UK/software_pedigreeSim.html and as Additional files 1, 2, 3 and 4 with this publication.

A genetic linkage map of tetraploid orchardgrass (Dactylis glomerata L.) and quantitative trait loci for heading date

Orchardgrass (Dactylis glomerata L.), or cocksfoot, is indigenous to Eurasia and northern Africa, but has been naturalized on nearly every continent and is one of the top perennial forage grasses grown worldwide. To improve the understanding of genetic architecture of orchardgrass and provide a template for heading date candidate gene search in this species, the goals of the present study were to construct a tetraploid orchardgrass genetic linkage map and identify quantitative trait loci associated with heading date. A combination of SSR markers derived from an orchardgrass EST library and AFLP markers were used to genotype an F1 population of 284 individuals derived from a very late heading Dactylis glomerata subsp. himalayensis parent and an early to mid-heading Dactylis glomerata subsp. aschersoniana parent. Two parental maps were constructed with 28 cosegregation groups and seven consensus linkage groups each, and homologous linkage groups were tied together by 38 bridging markers. Linkage group lengths varied from 98 to 187 cM, with an average distance between markers of 5.5 cM. All but two mapped SSR markers had homologies to physically mapped rice (Oryza sativa L.) genes, and six of the seven orchardgrass linkage groups were assigned based on this putative synteny with rice. Quantitative trait loci were detected for heading date on linkage groups 2, 5, and 6 in both parental maps, explaining between 12% and 24% of the variation.

Polyploidization increases meiotic recombination frequency in Arabidopsis: a close look at statistical modeling and data analysis

This paper is a response to Pecinka A, Fang W, Rehmsmeier M, Levy AA, Mittelsten Scheid, O: Polyploidization increases meiotic recombination frequency in Arabidopsis. BMC Biology 2011, 9:24.

See research article at http://www.biomedcentral.com/1741-7007/9/24

Statistical models for genetic mapping in polyploids: challenges and opportunities

Statistical methods for genetic mapping have well been developed for diploid species but are lagging in the more complex polyploids. The genetic mapping of polyploids, where genome number is higher than two, is complicated by uncertainty about the genotype-phenotype correspondence, inconsistent meiotic mechanisms, heterozygous genome structures, and increased allelic (action) and nonallelic (interaction) combinations. According to their meiotic configurations, polyploids can be classified as bivalent polyploids, in which only two chromosomes pair during meiosis at a time, and multivalent polyploids, where multiple chromosomes pair simultaneously. For some polyploids, these two types of pairing occur at the same time, leading to a mixed category. This chapter reviews several challenges due to the complexities of linkage analysis in polyploids and describes statistical models and algorithms that have been developed for linkage mapping based on their distinct meiotic characteristics. We discuss several issues that should be addressed to better understand the genome structure and organization of polyploids and the genetic architecture of complex traits for this unique group of plants.

Genotype calling in tetraploid species from bi-allelic marker data using mixture models

Background

Automated genotype calling in tetraploid species was until recently not possible, which hampered genetic analysis. Modern genotyping assays often produce two signals, one for each allele of a bi-allelic marker. While ample software is available to obtain genotypes (homozygous for either allele, or heterozygous) for diploid species from these signals, such software is not available for tetraploid species which may be scored as five alternative genotypes (aaaa, baaa, bbaa, bbba and bbbb; nulliplex to quadruplex).

Results

We present a novel algorithm, implemented in the R package fitTetra, to assign genotypes for bi-allelic markers to tetraploid samples from genotyping assays that produce intensity signals for both alleles. The algorithm is based on the fitting of several mixture models with five components, one for each of the five possible genotypes. The models have different numbers of parameters specifying the relation between the five component means, and some of them impose a constraint on the mixing proportions to conform to Hardy-Weinberg equilibrium (HWE) ratios. The software rejects markers that do not allow a reliable genotyping for the majority of the samples, and it assigns a missing score to samples that cannot be scored into one of the five possible genotypes with sufficient confidence.

Conclusions

We have validated the software with data of a collection of 224 potato varieties assayed with an Illumina GoldenGate™ 384 SNP array and shown that all SNPs with informative ratio distributions are fitted. Almost all fitted models appear to be correct based on visual inspection and comparison with diploid samples. When the collection of potato varieties is analyzed as if it were a population, almost all markers seem to be in Hardy-Weinberg equilibrium. The R package fitTetra is freely available under the GNU Public License from http://www.plantbreeding.wur.nl/UK/software_fitTetra.html and as Additional files with this article.

Em algorithm for mapping quantitative trait Loci in multivalent tetraploids

Multivalent tetraploids that include many plant species, such as potato, sugarcane, and rose, are of paramount importance to agricultural production and biological research. Quantitative trait locus (QTL) mapping in multivalent tetraploids is challenged by their unique cytogenetic properties, such as double reduction. We develop a statistical method for mapping multivalent tetraploid QTLs by considering these cytogenetic properties. This method is built in the mixture model-based framework and implemented with the EM algorithm. The method allows the simultaneous estimation of QTL positions, QTL effects, the chromosomal pairing factor, and the degree of double reduction as well as the assessment of the estimation precision of these parameters. We used simulated data to examine the statistical properties of the method and validate its utilization. The new method and its software will provide a useful tool for QTL mapping in multivalent tetraploids that undergo double reduction.

Multilocus tetrasomic linkage analysis using hidden Markov chain model

The availability of reliable genetic linkage maps is crucial for functional and evolutionary genomic analyses. Established theory and methods of genetic linkage analysis have made map construction a routine exercise in diploids. However, many evolutionarily, ecologically, and/or agronomically important species are autopolyploids, with autotetraploidy being a typical example. These species undergo much more complicated chromosomal segregation and recombination at meiosis than diploids. In addition, there is evidence of polyploidy-induced and highly dynamic changes in the structure of the genome. These polysomic characteristics indicate the inappropriateness of the theory and methods of linkage analysis in diploids for use in these species and a gap in the theory and methodology of tetraploid map construction. This paper presents a theoretical model and statistical framework for multilocus linkage analysis in autotetraploids for use with dominant and/or codominant DNA molecular markers. The theory and methods incorporate the essential features of allele segregation and recombination under tetrasomic inheritance and the major challenges in statistical modeling and marker data analysis. We validated the method and explored its statistical properties by intensive simulation study and demonstrated its utility by analysis of AFLP and SSR marker data from an outbred autotetraploid potato population.

Segregation models for disomic, tetrasomic and intermediate inheritance in tetraploids: a general procedure applied to Rorippa (yellow cress) microsatellite data

Tetraploid inheritance has two extremes: disomic in allotetraploids and tetrasomic in autotetraploids. The possibility of mixed, or intermediate, inheritance models has generally been neglected. These could well apply to newly formed hybrids or to diploidizing (auto)tetraploids. We present a simple likelihood-based approach that is able to incorporate disomic, tetrasomic, and intermediate inheritance models and estimates the double-reduction rate. Our model shows that inheritance of microsatellite markers in natural tetraploids of Rorippa amphibia and R. sylvestris is tetrasomic, confirming their autotetraploid origin. However, in F(1) hybrids inheritance was intermediate to disomic and tetrasomic inheritance. Apparently, in meiosis, chromosomes paired preferentially with the homolog from the same parental species, but not strictly so. Detected double-reduction rates were low. We tested the general applicability of our model, using published segregation data. In two cases, an intermediate inheritance model gave a better fit to the data than the tetrasomic model advocated by the authors. The existence of inheritance intermediate to disomic and tetrasomic has important implications for linkage mapping and population genetics and hence breeding programs of tetraploids. Methods that have been developed for either disomic or tetrasomic tetraploids may not be generally applicable, particularly in systems where hybridization is common.

Distinguishing major-gene from field resistance to late blight (Phytophthora infestans) of potato (Solanum tuberosum) and selecting for high levels of field resistance

Potato cultivar Stirling, which has a Solanum demissum derived R-gene and quantitative field resistance to late blight, was crossed with the susceptible cultivar Maris Piper to produce an F1 population from which three genotypes (94B13A29, 57 and 61) were backcrossed to Maris Piper. The F1 and backcross populations were assessed for resistance to simple race 1,4 (incompatible with Stirling's R-gene) and complex race 1,2,3,4,6,7 of Phytophthora infestans (compatible with R-gene) in whole plant glasshouse tests. The segregation results in the F1 generation with the simple race confirmed the presence of a single copy of the R-gene in Stirling, and the results with the complex race were consistent with Stirling having a high level of quantitatively inherited field resistance. Comparisons of the results with the simple and complex races apparently enabled F1 clones to be classified for the presence or absence of the R-gene and to be assessed for their level of quantitative field resistance. However, two out of the three backcrosses done to check classifications revealed unexpected findings: 94B13A29 had two copies of the R-gene as a result of double reduction, but was, as expected, susceptible to the complex race; and 94B13A57 had the R-gene (one copy) and it, and its offspring with the R-gene, had some resistance to the complex race, whereas those offspring without the R-gene were susceptible. Clone 94B13A61, as expected, lacked the R-gene and had moderate quantitative field resistance to both races. The implications are discussed for breeding potatoes with durable resistance to late blight.