Obesity has become a leading preventable cause of morbidity and mortality in many parts of the world. It is thought to originate from multiple genetic and environmental determinants. The aim of the current study was to introduce haplotype-based multi-locus stepwise regression (MSR) as a method to investigate combinations of unlinked single nucleotide polymorphisms (SNPs) for obesity phenotypes.2, SD = 4.1) or waist circumference  was assessed. Single SNP analyses were done by using linear regression adjusted for age, sex, and other covariates. Subsequently, MSR was applied to search for the ‘best’ SNP combinations. Combinations were selected according to specific AICc and p-value criteria. Model uncertainty was accounted for by a permutation test.In 2,122 healthy randomly selected men and women of the EPIC-Potsdam cohort, the association between 41 SNPs from 18 obesity-candidate genes and either body mass index  showed effects ranging from −1.70 (SE = 0.34) to 0.74 kg/m2 (SE = 0.21) per allele combination. We selected two six-SNP combinations on waist circumference  with an allele combination effect of −2.96 cm (SE = 0.76) at maximum. Additional adjustment for BMI revealed 15 three-SNP combinations . However, after carrying out the permutation test all SNP combinations lost significance indicating that the statistical associations might have occurred by chance.The strongest single SNP effects on BMI were found for TBC1D1 rs637797 , FTO rs9939609 , MC4R rs17700144 , and MC4R rs10871777 . All these SNPs showed similar effects on waist circumference. The two ‘best’ six-SNP combinations for BMI (global p-value = 3.45⋅10MSR provides a tool to search for risk-related SNP combinations of common traits or diseases. However, the search process does not always find meaningful SNP combinations in a dataset. Obesity is an increasing health problem worldwide that is associated with an increased risk of several common diseases including cardiovascular diseases, type 2 diabetes mellitus and certain cancers. The World Health Organization estimated that by 2008, 1.4 billion adults, 20 years and older, were overweight and from those more than 200 million men and nearly 300 million women were obese So far, there is limited data about the extent to which non-additive effects of genes, mostly described in terms of gene–gene interaction, will add to the inherited risk for obesity development. It is generally assumed that several loci could interactively contribute to common diseases or traits with higher magnitude of effects than the single variants. Resolving such combined effects is imperative to enable the identification of persons at high risk based on their genetic profile.In order to design a multi-locus based statistical tool to identify SNP combinations we extended the classical haplotype-based approach The aim of this study was to introduce an adapted version of the multi-locus stepwise regression (MSR) to combine SNP alleles from various chromosomes, i.e. unphased genotypes, in the way haplotypes are constructed Written informed consent was obtained from all study participants, and approval was given by the Ethical Committee of the Medical Association of the State of Brandenburg, Germany.2 ; women : 25.8±4.7, and men : 27.0±3.7). A random sample of 2,500 participants  was selected from those participants who had provided blood samples at baseline . From this representative ‘sub-cohort’ sample, participants with a history of diabetes, cancer, myocardial infarction, or stroke and those with missing relevant baseline information were excluded from the analysis. Hence, for the final analysis on anthropometric parameters data from 2,122 participants were used.The European Prospective Investigation into Cancer and Nutrition (EPIC)-Potsdam cohort study is part of a large multi-center European-wide cohort study Weight, height, and waist-circumference were measured by trained interviewers with standardized methods described elsewhere SNPs identified in genome wide association studies of body mass index or obesity , HSD11B1 (three SNPs), LEPR (three SNPs), MC4R (two SNPs), ABCC8 (six SNPs) and TBC1D1 (thirteen SNPs). The ATP-binding cassette transporter sub-family C member 8 gene (ABCC8), formerly known as pancreatic sulfonylurea receptor gene (SUR1), consists of ∼84 kilobases encompassing 39 exons that encode a protein of 1581 amino acids length Genotyping was performed before 2009 using TaqMan primers and technology  as described previously Genotype data of 41 SNPs in 18 candidate genes  distributed among 11 chromosomes were categorized into three groups . Single SNP effects on BMI and waist circumference were tested using a multiple linear regression model that assumed an additive effect by modeling the number of minor alleles .For estimating the effect of single SNPs and of the combined multiple SNPs (SNP combinations), the statistical model was adjusted for commonly known obesity-related factors such as sex, age at baseline, educational attainment , occupational activity , sports activity , smoking habits , alcohol intake , total daily energy (kJ/day), fat intake (g/day), and fruit and vegetable intake (g/day). Furthermore, the genetic effect on abdominal obesity was assessed by adjusting the models on waist circumference for BMI.Haplotype analysis was done for the six genes which comprised two or more SNPs . Haplotypes were estimated by the expectation-maximization (EM) algorithm SNP-sex interaction was tested by including a SNPxSEX interaction term in the adjusted models. The interaction p-values were determined by F test comparing the model including all covariates, SNP, sex, and the interaction term against the model excluding the interaction term.Statistical evaluation of SNP combinations was conducted by multi-locus stepwise regression (MSR) cc values showed an improvement and the p-value was below a certain threshold. The thresholds used for every step were 0.05, 0.01, 0.001, and 0.0001 for SNP combinations of two to five SNPs, respectively. The procedure can be repeated until a pre-defined number of maximum SNPs per combination is reached or no more SNP combinations fulfill the AICc and p-value criteria.The principle of MSR can be summarized as follows. The methods developed for haplotypes were used in the same way for unlinked genetic markers. SNP allele combinations (multi-locus marker) instead of haplotypes were tested. Allele combinations with a frequency less than 5% were rated as “rare” and subsequently pooled to avoid spurious associations due to low frequencies. Starting point was a pair-wise SNP association-test where all possible two-SNP allele combinations were tested for their association with the quantitative traits using multiple linear regression models. The individual probabilities of the SNP allele combinations consistent with the observed genotypes were estimated using the expectation-maximization (EM) algorithm The effect of each single allele combination included in the best selected SNP combinations were assessed by comparing the results from the multiple linear regression models including each allele combination and all covariates against the model including only covariates. We used the F test to determine the statistical significance for each single allele combination, separately.2 means a prediction of a 1.70 kg/m2 lower average of BMI for individuals which were heterozygous for that allele combination compared to individuals which had a genotype which was not comparable to that allele combination. If an individual is homozygous for that allele combination, then the expected mean of BMI would be 2⋅1.70 = 3.40 kg/m2 lower than individuals carrying none of this allele combination.Each allele combination is characterized by the combination of major (coded as 1) or minor alleles (coded as 2) of the included SNPs (loci). For instance, the allele combination 112111 (the first six-SNP combination derived from MSR for BMI) consists at the first given two loci of the major alleles, followed by one minor allele and at the last given three loci of the major alleles. All individual genotypes could be assigned with pairs of such allele combinations weighted by their estimated frequencies. Each individual genotype is compatible to none, one (heterozygous), or pairs (homozygous) of such allele combinations. The beta-coefficient for that allele combination of −1.70 kg/mThe effect of model uncertainty was assessed by using a resampling method Using the permutation approach we derived the distribution of the test statistic under the null hypothesis by random permutations of the trait values and holding the non-genetic covariates constant relative to the trait values. In each permutation the SNP combinations remained unchanged. In this permutation approach the phenotype-covariate-association was held constant by keeping the covariates for each trait value. We focused on the relationship between the phenotype and the SNP combinations but maintained the relationship within the SNP combinations. These permutation samples are assumed to represent the distribution under the null hypothesis. The permutation-based p-value was calculated as proportion of the lowest p-value of each permutation sample less than the observed p-value.The effect of random selection of the best SNP combinations was assessed by using 200 permutations and running the MSR for each permutation. The permutations could result in SNP combinations with different lengths, i.e. number of combined SNP alleles, and different SNPs. The lowest p-value of each permutation was stored, and results were compared based on the original sample with the distribution of the lowest p-values derived by the permutation approach.We evaluated the chance of “random selection” by a small simulation study (type I error). Assuming no genetic effect we simulated 41 unlinked SNPs with equal minor allele frequencies of 0.2 for 2,156 individuals. Each individual was assigned to a random phenotype value taken from a normal distribution with mean = 25.9 and standard deviation = 4.1, which equaled the average BMI value in our study. This procedure was repeated 1,000 times and within each replication we applied MSR. For simplicity and since the simulation study was small, we only used the p-value for interpretation.The Statistical Software R (version 2.14.0) 2, SD = 3.4) and women , were slightly above the threshold for “normal” weight. The mean waist circumference was higher for men  than for women . Further, the participants were predominantly highly educated and less active. Most of the participants were nonsmokers (48% never and 32% former smoker) and drank less than 40 g/day alcohol (81%). No alcohol was consumed by 3% of the participants. Mean energy , fat,  and fruit and vegetable intake  differed between men and women.A description of baseline characteristics of the study sample for this analysis is given in All SNPs had a minor allele frequency (MAF) of at least 5% and passed the Hardy-Weinberg equilibrium (HWE) test . The average missing genotype proportion was 2.1%. The obtained MAFs were consistent with those reported in the HapMap CEU data of European descent.rs17700144 = 1.11, SE = 0.38, p = 0.003 and ßrs10871777 = 0.91, SE = 0.36, p = 0.012) showed also an association with waist circumference. Nominal significant effect on waist circumference was observed for IGF1 rs1520220  and a similar effect for PPARG rs1801282 . Nominal significant effect on waist circumference adjusted for BMI was found for ABCC8 rs10832786 . All single SNP associations did not withstand corrections for multiple testing , FTO rs9939609 , MC4R rs17700144 , and MC4R rs10871777 . TBC1D1 rs637797  and MC4R SNPs  on BMI.We applied haplotype analysis on SNPs in LEPR, HSD11B1, TBC1D1, FABP2, ABCC8, and MC4R. None of the tested haplotypes showed robust effects with exception of MC4R on BMI and waist circumference which showed the same effect as the single-loci analysis –3. For eFive SNP-sex interaction terms resulted in p-values lower than 0.05, but would not survive a correction for multiple testing : PPARG r–2 (two-SNP combinations) to 5.43⋅10–4 (four-SNP combinations) and 7.71⋅10–6 (six-SNP combinations). Decreased p-values were also observed for the two search processes on waist circumference . FTO rs9939609 was also present in the two selected SNP combinations related to waist circumference but not in the best SNP combinations for waist circumference adjusted for BMI.For the best six-SNP combination identified for BMI, the four most common allele combinations showed frequencies of 0.168, 0.131, 0.064, and 0.062, respectively . The sec2 (SE = 0.34) per allele combination to 0.74 kg/m2 (SE = 0.21) (2 (SE = 0.15). The effects on waist circumference not adjusted for BMI were stronger compared to the effects on waist circumference adjusted for BMI. The strongest effect of allele combinations on waist circumference was −2.96 cm (SE = 0.76) and on waist circumference adjusted for BMI was −1.26 cm (SE = 0.36) (The effect sizes for BMI ranged from −1.70 kg/m = 0.21) . Compare = 0.36) .However, permutation testing to evaluate the effect of random selection of the MSR under the null hypothesis of no effect showed that all of the constructed permutation p-values for each phenotype were higher than the commonly used threshold of 5% . All ide–12 to 3.5⋅10–2 with 5% percentile of 5.6⋅10–9 and median p-value of 4.9⋅10–6.On average 5% of the single SNP p-values from each MSR replication were lower than 0.05 indicating that the simulation process might have been successful because we modeled no genetic effects. The minimal p-value of each MSR run ranged from 1.5⋅102 for BMI and −0.85 cm for waist circumference per minor allele of TBC1D1 rs637797, respectively. Furthermore, we observed no haplotype effects in five genes  and in MC4R two haplotypes showed similar effects as found in the single SNP tests.We applied a multi-locus stepwise regression approach on 41 SNPs from known candidate gene loci for obesity traits to find trait-related SNP combinations in a population-based study sample, consisting of >2000 middle-aged participants of the EPIC-Potsdam cohort. As starting point we found some candidate gene SNPs to be nominally associated with BMI, waist circumference, and waist circumference adjusted for BMI including for instance FTO rs9939609 . Thus, this allele combination showed a remarkably large effect.Following our hypothesis that combinations of candidate gene SNPs might explain a larger proportion of the heritability of obesity-related anthropometric phenotypes we applied multi-locus stepwise regression (MSR) to identify trait-related SNP combinations. MSR revealed several low-frequency allele combinations (frequencies range: 5% to 23%) for BMI and waist circumference within those SNP combinations which showed a considerably higher impact on the obesity-related anthropometric phenotypes compared to single SNP associations. For example, the allele combination comprising the major alleles of HSD11B1 rs4844880, TBC1D1 rs10009706, IGF1 rs1520220, FTO rs9939609, MC4R rs17700144, and the minor allele of FABP2 rs1799883 was associated with a decrease of −1.70 kg/mThis study is hypothesis-driven since it utilizes prior information from genome-wide association, candidate gene and other studies. Given the high number of null hypotheses tested in the MSR, excessive adjusting of p-values for multiplicity could seriously hinder finding relevant genetic variants. However, too low restrictions within the stepwise search process would result in a magnitude of SNP combinations more likely to be false-positive. As Curtin et al. (2010) recommended for their algorithm implemented in hapConstructor we used stringent significance thresholds in each step of the MSR to focus on final SNP combinations which may not fail multiple testing corrections k possible genotypes. Depending on the minor allele frequencies many of the multi-SNP genotypes could be rare (frequencies <0.05). Therefore, the usage of genotypes is not as useful for combining many unlinked SNPs as the use of alleles.Curtin et al. (2010) used the hapConstructor approach c criterion as measurement of goodness–of-fit, thus taking account of the model complexity. We applied the MSR to find allele combinations associated with continuous traits instead using SNP genotypes associated with disease as done by Curtin et al. (2010).In some respect, the MSR approach is similar to the hapConstructor approach Compared to the observed genotypes, allele combinations were inferred in each step of the MSR. Therefore, interpretations of such allele combinations should only be done on the background of all included SNPs and with care.combinations = , for instance if n = 6 we get 5,358,577 possible SNP combinations. However, these combinations are not independent because each SNP will be part of different SNP combinations.The correct number of independent test statistics for the application of Bonferroni- or other correction methods (e.g. FDR) while running MSR is unknown. The number of independent tests for correction lies somewhere between the number of actual conducted tests and the number of all possible tests up to n-SNP combinations nTherefore, the model uncertainty of the selection strategy was evaluated by using a resampling method: The distribution under the null hypothesis was simulated by 200 permutations of the trait values holding the relationship between the phenotype and non-genetic covariates constant. It was shown that each of the permuted p-values of the best SNP combination were higher than the arbitrary significance level of 5% indicating that our result might have occurred by chance due to the high number of possible SNP combinations. Our finding points to a serious problem in a combination analysis. The selection strategy of unlinked marker combinations can yield significant false-positive effects due to the high number of possible combinations. The sole solution to overcome this problem is an independent replication in a second study sample, which was not available for our study.The small simulation study showed that under the assumption of no genetic effect the simulated p-values are in the range of the p-values of the observed SNP combinations. This result also suggests that our selected SNP combinations are very likely to be a random finding. More intensive investigations on the feasibility and power of haplotype-based stepwise regression models for selecting SNP combinations of unlinked SNPs are needed.men = 26.6, BMIwomen = 25.5). The genetic effect was assessed on a continuous scale of obesity-related phenotypes and not restricted to obese individuals which may have attenuate our empirical results, i.e. in another study sample enriched with extreme phenotypes, associations might have been stronger.Our study population consisted of a random sample representative of the general population of a distinct area. On average study participants were slightly overweight  with obesity-related measurements (BMI and waist circumference) did not withstand multiple testing correction although some single variants and SNP combinations showed meaningful effects. We conclude that the use of systematic search procedures like MSR requires careful consideration of the search process in order to minimize the chance for false-positive findings. Resampling methods can be used to investigate such model uncertainties.Figure S12 values were generated using Haploview version 4.2 :263-5). Standard Color Scheme for the LD plots for r2 were used (for more information see Haploview documentation).LD plot for ABCC8 gene of the EPIC Potsdam subsample . Disequilibrium coefficient r(PDF)Click here for additional data file.Figure S22 values were generated using Haploview version 4.2 :263-5). Standard Color Scheme for the LD plots for r2 were used (for more information see Haploview documentation).LD plot for TBC1D1 gene of the EPIC Potsdam subsample . Disequilibrium coefficient r(PDF)Click here for additional data file.Table S12) in the EPIC-Potsdam subsample  with adjustment for sex, age at baseline, educational attainment, occupational activity, sports activity, smoking habits, alcohol intake, energy intake, fat intake, and fruit and vegetable intake.Single haplotype analysis on body-mass index (kg/m(PDF)Click here for additional data file.Table S2Single haplotype analysis on waist circumference (cm) in the EPIC-Potsdam subsample  with adjustment for sex, age at baseline, educational attainment, occupational activity, sports activity, smoking habits, alcohol intake, energy intake, fat intake, and fruit and vegetable intake.(PDF)Click here for additional data file.Table S32) in the EPIC-Potsdam subsample  with adjustment for sex, age at baseline, educational attainment, occupational activity, sports activity, smoking habits, alcohol intake, energy intake, fat intake, and fruit and vegetable intake.Single haplotype analysis on waist circumference (cm) adjusted for body-mass index (kg/m(PDF)Click here for additional data file.Table S4Evaluation of SNP-gender interaction in 2,122 European middle-aged men and women (random population sample).(PDF)Click here for additional data file.Table S52), waist circumference  not adjusted and adjusted for BMI in the EPIC-Potsdam subsample .Mean p-values of selected SNP patterns in each step of the multi-locus stepwise regression (MSR) with 41 SNPs on Body-mass index Click here for additional data file.Table S62) in the EPIC-Potsdam subsample . Starting with SNP-pairs one SNP at a time was added to the ‘best’ patterns in the interim step. Selection criterion in every step was a decrease of corrected AIC  and a global p value below a given threshold . SNP numbers correspond to identification number in Result of Multi-locus stepwise regression with 41 SNPs on BMI (kg/m of the main text.(PDF)Click here for additional data file.Table S7Result of Multi-locus stepwise regression with 41 SNPs on waist circumference (cm) in the EPIC-Potsdam subsample . Starting with SNP-pairs one SNP at a time was added to the ‘best’ patterns in the interim step. Selection criterion in every step was a decrease of corrected AIC  and a global p value below a given threshold , where i denote the number of simultaneously analyzed SNPs in each step). SNP numbers correspond to identification number in  of the main text.(PDF)Click here for additional data file.Table S8Result of Multi-locus stepwise regression with 41 SNPs on waist circumference (cm) adjusted for BMI in the EPIC-Potsdam subsample . Starting with SNP-pairs one SNP at a time was added to the ‘best’ patterns in the interim step. Selection criterion in every step was a decrease of corrected AIC  and a global p value below a given threshold , where i denote the number of simultaneously analyzed SNPs in each step). SNP numbers correspond to identification number in  of the main text.(PDF)Click here for additional data file.Table S9All other associations of the MSR-selected SNP combinations with obesity-related traits  in 2,122 European middle-aged men and women (random population sample).(PDF)Click here for additional data file.Text S1Physiological mechanisms of selected genes in our study.(PDF)Click here for additional data file.


Sorghum bicolor (L.) Moench] underlying economically important traits, we analyzed the genetic structure and linkage disequilibrium in a sorghum mini core collection of 242 landraces with 13,390 single-nucleotide polymorphims. The single-nucleotide polymorphisms were produced using a highly multiplexed genotyping-by-sequencing methodology. Genetic structure was established using principal component, Neighbor-Joining phylogenetic, and Bayesian cluster analyses. These analyses indicated that the mini-core collection was structured along both geographic origin and sorghum race classification. Examples of the former were accessions from Southern Africa, East Asia, and Yemen. Examples of the latter were caudatums with widespread geographical distribution, durras from India, and guineas from West Africa. Race bicolor, the most primitive and the least clearly defined sorghum race, clustered among other races and formed only one clear bicolor-centric cluster. Genome-wide linkage disequilibrium analyses showed linkage disequilibrium decayed, on average, within 10−30 kb, whereas the short arm of SBI-06 contained a linkage disequilibrium block of 20.33 Mb, confirming a previous report of low recombination on this chromosome arm. Four smaller but equally significant linkage disequilibrium blocks of 3.5−35.5 kb were detected on chromosomes 1, 2, 9, and 10. We examined the genes encoded within each block to provide a first look at candidates such as homologs of GS3 and FT that may indicate a selective sweep during sorghum domestication.To facilitate the mapping of genes in sorghum [ Sorghum bicolor (L.) Moench] is a grass of the steppes and savannas of Africa that was domesticated 3000−5000 years ago and spread to diverse climates and geographical locations worldwide (Sorghum [orldwide . The speorldwide . Under torldwide . Guinea orldwide . Kafir iorldwide . The durGenetic structure for populations composed of different races affects the efficiency of association (or linkage disequilibrium—LD) mapping. Early studies of population structure in sorghum were focused on differentiating the races based on a limited number of genetic markers. Using restriction fragment length polymorphism markers, Population genetic structure is a critical factor influencing the outcome of association or LD mapping . In sorgWith the advent of genome-wide, sequenced-based genotyping in sorghum, precise and accurate estimations of population structure and LD across the genome are now attainable. Our goal in this research was to use genotyping-by-sequencing technology to reexamine population structure and LD of a mini-core collection developed from the International Crops Research Institute for the Semi-Arid Tropics (ICRISAT) sorghum core collection. This mini-core collection of 242 accessions is composed entirely of landraces from 57 countries, which were selected based on 11 qualitative and 10 quantitative traits . Adapters containing 4-bp identifier tags were ligated onto the FseI-cut end of the resulting fragments, and following adapter ligation, the samples were pooled in groups of 48. Pools were precipitated, sheared to 200−600 bp using a Biorupter sonicator , size selected (150−250 bp) by agarose gel electrophoresis and subsequently purified by gel extraction . Samples were then treated with Bst DNA Polymerase (New England BioLabs) to repair the ends, A-tails were added to the sheared ends using Klenow 3′-5′ exonuclease (New England BioLabs), and the samples blunt-ended using a Quick Blunting Kit (New England BioLabs). After each of the aforementioned steps, a purification step (QIAGEN) was performed. Following end-repair and A-tailing, a second set of adapters was ligated onto the 3′ end of the product with a 5′ identification adapter and a 3′ general adapter. Ligation reactions were cleaned using AMPure XP Reagent  to remove excess adapter. Subsequently, products were enriched by PCR amplification using Phusion High-Fidelity DNA Polymerase (New England BioLabs) from the adapters using one standard and one biotinylated primer. The desired products were isolated using magnetic Dynabeads M-280 Streptavidin beads , and the captured double-stranded DNA was denatured at 98° to form single stranded products. The nonbiotinylated product was carried forward through a second round of PCR. The final PCR resulted in double-stranded products that contained the end sequences necessary for bridge-amplification, and samples at 10 nM concentration were submitted for 78-bp single-end sequencing on an Illumina GAIIx . For each PCR step, a high-fidelity DNA polymerase with 3ʹ−5ʹ proofreading ability and high processivity was used to reduce the introduction of sequencing errors. In addition, the number of PCR cycles was kept as low as possible to obtain the desired quantity of PCR product while limiting the introduction of PCR errors.DNA was isolated from either germinating seedlings or frozen leaf tissues using the FastDNA Spin Kit  and quantified using a fluorometric assay . DNA purity was assessed by obtaining 260/280 ratios and confirmation of an absorbance maximum at 260 nm using the NanoDrop1000 instrument . SNPs were generated using high-throughput sequence analysis of DNA templates from specific sites targeted by restriction endonucleases .Base call conversion and assignment of quality scores was performed using Illumina’s CASAVA_v1.7 software. Sequences were trimmed, sorted by individual 4-bp multiplex identifier (MID) tags, and filtered for 100% identity of the individual MID tag and partial restriction site sequence. The processed reads for each individual accession were aligned to the sorghum genome sequence . Followihttp://www.doka.ch/Excel3Dscatterplot.htm) using the PC data output. STRUCTURE 2.3 . Local LD analysis on SBI-06 used the full matrix option in TASSEL. The critical value of r2 was the conventional 0.1  was searched for annotated genes at www.phytozome.net.PCA was performed using the package prcomp . SNP datFseI. This enzyme recognizes the GC-rich sequence GGCCGGCC that is found in or near many nuclear genes but is not found in the sorghum chloroplast genome. Following template library preparation, 48 accessions, each containing a unique 4-bp MID tag, were pooled/lane on the Illumina GAIIx. An average of 504,000 78-bp sequences was obtained from each accession that contained the expected FseI partial restriction site and 4-bp MID tag. Approximately 91% of the reads from each accession mapped uniquely to the sorghum genome reference sequence. The sorghum genome contains ∼23,000 FseI sites based on in silico digestion of the reference BTx623 sequence. Therefore, if every recognition site were restricted with FseI, ∼46,000 reads  would be expected. However, because FseI is sensitive to methylation, it will only restrict nonmethylated recognition sequences. On average, 22,000 unique reads/genotype were obtained representing ∼1.58Mbp of the sorghum genome. This result was not surprising given the fact that a large percentage (∼50%) of sequences that flank FseI sites across the sorghum genome are repetitive and a subset of these will be methylated.The 242 accessions from the ICRISAT mini-core collection were genotyped using a genotyping-by-sequencing approach. Sequencing templates were generated using the methylation-sensitive enzyme i.e., reference allele, alternative allele(s) or heterozygous alleles) be obtained from at least 25% of the 242 genotypes analyzed before imputation of missing data. To serve as an internal standard for estimating the extent to which sequencing errors would falsely predict a SNP, template was prepared from the reference genotype, BTx623, and included in multiple lanes on the Illumina flow cell. Ninety-five percent of the BTx623 reads mapped uniquely to the BTx623 reference genome and an average of 115 SNPs were detected across the 8 BTx623 samples (range of 76 to 173) analyzed compared with an average of 5507 SNPs identified in the genotypes representing the mini core collection (data not shown). Only 102 SNPs identified in the mini core collection were present in the BTx623 samples indicating that the SNP calling parameters utilized resulted in a false-positive SNP call less than 1% of the time . Taken together, these results indicate that the library preparation, mapping and SNP detection parameters utilized in the present study identified a high quality set of SNPs for downstream use.The reads that uniquely mapped to the reference genome were used to identify SNPs using the SNP Detection Tool in the CLC Bio Workbench. The number of SNPs obtained per accession following mapping to the BTx623 sorghum reference sequence ranged from a high of 13,099 to a low of 2093 . Comparison of SNPs among the 242 accessions identified 76,138 SNPs in at least 2 of the 242 accessions examined. Several steps were taken to ensure that a high-quality set of SNPs was obtained. At each PCR step during template preparation, a high-fidelity, proof-reading polymerase with high processivity was used, and the number of cycles was limited to reduce incorporation of sequencing errors during PCR amplification. During SNP calling in the CLC Bio Workbench, a minimum of six reads was required to cover the position of the SNP to be called. We required that base calls . As expected a greater density of SNPs/Mbp were found in the euchromatic arms of each chromosome where gene density is greatest, and fewer SNPs/Mbp were found in the repeat-rich pericentromeric heterochromatic region of each chromosome consistent with these regions having greater levels of methylation and reduced rates of recombination . Analysis of the position of each SNP shows the majority (∼72%) are within 1 kbp of an adjacent SNP .After removal of SNPs that mapped to super_contigs, those that did not map near an The 13,390 SNPs were coded in numbers and used to run prcomp in R. To determine the number of PCs to use in clustering the mini-core accessions, a scree plot was generated in which the proportion  of an individual PC’s contribution to total variation was plotted against the number of PCs . The chaThe first three PCs were plotted in 3D to best capture the distribution of all accessions . The accTable S2). The majority of the remaining PGs were formed based on race. The most intriguing were guineas, which were distributed among three PGs. Guineas from West Africa  formed PG6 together with hybrid races also from the region. PG10 contained guinea landraces from the East African countries of Tanzania, Mozambique, and Malawi. The smallest guinea cluster was PG9 with guineas from India. Race caudatum formed two groups, PG7 and PG8. PG7, the major cluster, included caudatums from countries in Africa, Asia, and the United States as well as guinea-caudatums and durra-caudatums. The smaller cluster, PG8, also contained caudatums, guinea-caudatums and durra-caudatums but the accessions in this cluster were solely from Africa. PG1 was a loose cluster of mostly bicolors and caudatum-bicolors. The only group that neither followed race nor geography was PG5, comprised of various races from Africa, the US, Mexico, Syria, and Yemen. Fourteen accessions were not considered clustered  in the ICRISAT mini-core collection with the 13,390 SNP markers used for PCA and phylogenetic analysis. To determine the appropriate value of k, STRUCTURE was performed 10 times with assumed k from 2-16. The k value where the posterior probability (ln P(D)) began to plateau was selected as the true k  dropped to less than −2,200,000  6 included PG3 and half of PG1. When k = 12, PG5-1 and PG2 were in one SG, whereas they were separate SGs at k = 11. Setting k at 13 divided PG7 into two SGs—this is not ideal considering that PG7 was a tight group based on both PCA and phylogenetic analysis . In general, however, results from STRUCTURE were in agreement with those from PCA and phylogenetic analysis. Our results from the three clustering methods demonstrate that bicolor was not sufficiently divergent from the other races to form one inclusive group. However, the other four races were genetically distinct. This is in agreement with conclusions reached in previous studies  used in this study was the conventional 0.1  at 30–50 kb across all 10 chromosomes. This result was corroborated by the fact that after visually scanning through the chromosome LD plots, no large LD blocks in euchromatic regions were found other than one on SBI-06 (see below). The results agree with r2 within 50 kb from 0.18 to 0.13.The average r SBI-02 . On SBI-r2 values between these regions in each chromosome . The LD block detected on SBI-10 was 4.59 Mb with an r2 of 1.00 and a p value of 3.40 × 10−5. It was located between 44.18–48.78 Mbp on SBI-10 and flanked by five SNPs on one side and four on the other (data not shown).In addition to the largest LD block spanning the SBI-06 heterochromatic region, two smaller LD blocks were found, one on SBI-06 and the second on SBI-10. The smaller LD block on SBI-06 was 160.79 kb with an average i.e., selection may create a large LD block that contains domestication-related genes  homologous to rice GS3 with 71% identity and 76% similarity based on GenBank BLASTP results  and rice FT homolog, Hd3a . FT promotes flowering in Arabidopsis in conjunction with LFY. Mutations in FT delay flowering , random drift, or gene flow has taken place within a given race. Two prominent LD blocks were found on SBI-01 and SBI-02, and in both cases the LD block was specific to race guinea.Ghd7 and CONSTANS in the CCT motif, with the closest rice homolog (Os03g0139500) being 63% identical and 75% similar to Sb01g047650. The other sorghum gene in this genomic region encoded an unknown protein. It is possible that the CCT-motif encoding protein in this region was under selection during domestication of race guinea, but why LD was not detected in the other races is unclear. However, because CCT-domain proteins regulate photoperiodic flowering, vernalization, circadian rhythms, and light signaling, it is possible that an allelic variant of this protein was under selection during the domestication of guineas in Western Africa, although other factors such as genetic drift cannot be excluded.The LD block on SBI-01 found in guinea was bifurcated between chr1_70739511 and chr1_70752339 in bicolor, caudatum, durra, and kafir . A searcSbHox14 (Sb02g037560) located between 71,846,508 and 71,847,734 bp and a sugar transporter. SbHox14 was 66% identical and 75% similar to barley Vrs1 (HvHox1) which is expressed predominantly in the immature spike, the equivalent of a panicle in sorghum   indicated that the collection was structured along both geographic origin and race. Accessions of different races from southern Africa tended to be more similar to each other, as were those from East Asia. Caudatums from different countries were clustered, which was the strongest example of population structured based on race. Guineas from West Africa and durras from India were clustered by race and origin. On the other hand, race bicolor, the most primitive and ancestral of the cultivated races, was found largely distributed among the other four races. This complements reports that sorghum domestication was the product of independent domestication events . SbHox14 is similar to barley Vrs1, in which a mutation converts two-rowed barley into six-rowed barley, again affecting panicle (spike) morphology. Although purely correlative, these smaller LD regions encode genes related to photosensitivity and panicle/grain morphology. As the sorghum races were established on panicle architecture and geographical origin, the relationship between circadian clock-associated and panicle morphology genes and regions of LD across the sorghum genome warrant further investigation.Genome-wide LD analysis showed that LD decayed within 10−30 kb, close to that previously reported (


However, both read length and contig depth are highly variable and thus raise doubt about simple methods of SNP frequency estimation.Next-generation sequencing methods have contributed to rapid progress in the fields of genomics and population genetics. Using this high-throughput and cost-effective technology, a number of studies have estimated single nucleotide polymorphism (SNP) frequency by calculating the mean number of SNPs per unit sequence length (Ambystoma tigrinum tigrinum) transcriptome, then constructed analytical models to estimate SNP frequency. The model which considered only contig length  was evaluated with very poor likelihood. Our most robust model considered read depth as well as contig length, and was 7.5 × 1055 times more likely than the length-only model. Using this novel modeling approach, we estimated SNP frequency in protein-coding (mRNA) and non-coding transcripts . We found little difference in SNP frequency in the contigs, but we found a trend of a higher frequency of SNPs in long contigs representing non-coding transcripts relative to protein-coding transcripts. These results support the hypothesis that long non-coding transcripts are less conserved than long protein-coding transcripts.We used 454 pyrosequencing to identify 2,980 putative SNPs in the eastern tiger salamander  can be a powerful tool for identifying selection on specific functional sequence groups by comparing the frequency and distribution of polymorphisms.A modeling approach ( Single nucleotide polymorphism (SNP) profiles can be used to infer the intensity of selection by comparing the frequency and distribution of SNPs of two or more sequence regions, such as genic vs. intergenic regions, exonic vs. intronic vs. untranslated regions, autosomes vs. sex chromosomes, and particular gene groups with specific functions [Mutations and subsequent polymorphisms are important not only for identifying influences on genetic disease ,2, but aunctions -9.e.g., among functional gene groups, among chromosomes, among populations), large samples of similar size are desirable. For example, the International HapMap Consortium [e.g., [To properly compare SNP frequency among groups  has contributed to the rapid development of SNPs in non-model but important wild species [e.g., mean SNP/kb), thus ignoring bias due to contig depth. One of our goals was to determine if such length-only-methods can effectively estimate SNP frequency from next-gen datasets or if more sophisticated approaches are needed.The advent of high-throughput sequencing technologies  and transfer RNA (tRNA). In general, non-coding transcripts are expected to be less conserved  than protein-coding regions because selection may eliminate deleterious mutations from functional sequences [A second goal was to compare SNP frequencies in protein-coding transcripts equences ,16. Howeequences . This maequences -20, the A. t. tigrinum) using 454 pyrosequencing and then analyzed the distribution and frequency of SNPs. We studied the influence of contig depth on the estimation of SNP frequencies, and we tested the hypothesis that non-coding transcripts are less conserved than protein-coding transcripts by comparing their SNP frequencies.One of our ongoing research programs is focused on evolutionary and ecological genomics of tiger salamanders -24. For SfiI to remove excess primers and subsequently purified with the QIAquick PCR purification kit (Qiagen). Approximately 5 μg of each cDNA library was sequenced using a 454 Genome Sequencer with FLX Titanium chemistry (454 Life Sciences). Briefly, the cDNA fragments were sheared via nebulization, hybridized to DNA capture beads, and amplified by emulsion-based PCR. Molecular barcodes with multiplex identifiers  were applied to specifically tag each cDNA library in our pooled sequencing run.We captured and sampled seven tiger salamanders at the Purdue Wildlife Area  consistent with IACUC 1203000614 and in accordance with guidelines by the American Society of Ichthyologists and Herpetologists and by the American Veterinary Medical Association (June 2007). RNA was extracted from gill, lung, skin and spleen using TRIzol® reagent (Invitrogen). We constructed a total of 19 tissue-specific cDNA libraries from these RNA samples using the ClonTech SMART cDNA synthesis kit, utilizing a modified poly-T primer . Each doWe pooled all sequences from seven individuals in PCAP  and asseWe divided all contigs into categories of protein-coding and non-coding transcripts using the program CPC . CPC empA priori, we predicted that SNP frequency would positively associate with both factors. We employed the information-theoretic approach suggested by Akaike [Our first goal was to evaluate the effects of contig length and depth on the estimated frequency of SNPs. y Akaike  and extey Akaike  to evaluwi). We repeated construction of candidate models and model selection for the number of transitions (Ti) and transversions (Tv) as the response variables, as well as for the number of SNPs in a contig. For ease of interpretation of the relative effect of each variable, we plotted estimates of the relationships between length or depth of contigs and the number of SNPs found in protein-coding and non-coding transcripts.Four candidate models were fitted using regression models based on a negative binomial distribution. To assess the relative fit of each model, we compared the Akaike’s Information Criteria (AIC), with better fitting models having lower AIC . The rele.g., repetitive elements, short reads, etc.) we assembled the remaining 273,501 high-quality sequences into 51,391 contigs. Among contigs with at least 10 sequence reads, 4,552 and 3,181 contigs were ≥100 and ≥501 bp in length, respectively ; the pattern was reversed in the “long” dataset of ≥501-bp contigs where protein-coding transcripts were more common . A total of 2,980 SNPs that passed our stringent scoring criteria were detected in the “all” contig dataset; of these, 2,515 (84%) were found in the “long” dataset  and non-coding transcripts ; all of these associations were significant (P < 0.001).The correlation results were qualitatively similar between the “all”  and transversions (Tv) (Table wi (relative plausability or weight of each model) of the global models for estimating the frequency of SNPs, Ti, and Tv, ranged from 0.68 to 0.99. In sharp contrast, those of the commonly-employed “length-but-no-depth” models ranged from 6.7 × 10-24 to 1.2 × 10-56. Our best-fit model predicting SNP frequency was 7.5 × 1055 (= 0.904/(1.2 × 10-56)) times more likely than the length-but-no-depth model (Table To estimate SNP frequency in the ‘long’ (≥ 501 bp and ≥10 reads) contig dataset, we used the negative binomial distribution to construct four regression models that considered contig length, contig depth, and the category of transcript (protein-coding and non-coding transcript). The most robust model of SNP frequency was the global model that used all of these variables; similarly, the global models were most robust for estimating the frequency of transitions (v) Table . The wi P = 0.08) but this trend was not apparent in the “all” contig dataset  of the models, because wi is best interpreted as the relative likelihood or probability of the model [55 times more likely than the model of length and transcript type .Our modeling approach can be used as a tool for inferring the intensity of selection on different categories of genes or genomic regions by comparing the SNP frequency among categories. Similar approaches have been developed for nucleotide diversity parameters such as beta (β) . We studet al.[In the current analyses, we included the dummy variable C/NC in all candidate models to test for differences in SNP frequencies between protein-coding and non-coding transcripts. In the long dataset with 10 or more reads we found a negative coefficient of C/NC, implying SNP frequency in non-coding transcripts is higher than that in protein-coding transcripts. Our use of a poly-dT primer for cDNA synthesis may have biased our representation of the transcriptome, but a similar pattern has been reported in human genomic DNA. Zhao et al. investiget al.,16. Howeet al.. Althouget al. from pooet al.,37.A. t. tigrinum. These results are consistent with the observation that different classes of functional, non-coding RNAs  evolve at different rates in humans and mice, with faster evolution in longer functional non-coding RNAs [vs. non-coding) can be limited by the number of reference genomes available for comparison [e.g., no complete genome sequence is available) and as of yet has no close relatives with sequenced genomes, so this complicates the identification of orthologous sequences and protein homologs used in categorical (coding/non-coding) transcript assignments.Our data indicate there is considerable variation in SNP frequency across transcribed sequences in ing RNAs . That saing RNAs  to parseing RNAs ,29, but e.g., 5’ or 3’ UTR) that occurs in a true protein-coding transcript. Such sequences will be erroneously categorized as non-coding transcripts due to poor ORF-related score, obfuscating SNP frequency differences between protein-coding and non-coding transcripts.A second source of bias may be the short read sequences generated by next generation sequencing. For example, some of our sequences may be derived from short sections of an untranslated region  [vs. non-coding transcripts when all contigs were considered, but we did observe a trend of greater SNP density in non-coding transcripts compared to coding transcripts when only long contigs were considered. This observation provides indirect evidence that “long” non-coding transcripts have relatively fast rates of molecular evolution.Finally, there may be true biological differences in SNP frequency between protein-coding and non-coding transcripts or within a transcript category—particularly when considering “long” transcripts. Recent analyses have revealed that even transcripts which do not encode proteins may have important functions and some of them are conserved to maintain functional domains and structures ,38-40. Moss taxa ,41. Thisfor Air) ,43. Suchthe standardized number of SNPs for a given contig length and depth instead of the current “standard” of SNPs per unit length . For example, polymorphism frequency might be reported as the number of SNPs in a 1 kb-long transcript with coverage depth of 10 reads. Even longer and deeper contigs will become more common as sequencing technology continues to advance (so it is currently fruitless to recommend a particular read depth), but authors should at least provide readers with information about contig depth as well as length when reporting SNP frequencies.Our transcriptome results revealed that estimates of polymorphism frequency are affected by both length and depth of contigs, such that simply dividing the total number of SNPs by the sequence length produces biased frequency estimates. In next-generation sequencing studies where reads vary dramatically among contigs, we propose that estimates of SNP frequency should be presented as i.e., using multiple model construction and model selection approaches) can be a powerful tool for identifying selection on specific functional sequence groups by comparing the frequency and distribution of polymorphisms.Our modeling approach revealed that long non-coding transcripts have marginally higher SNP frequencies than protein-coding transcripts. Future models that consider more variables may further refine our ability to estimate SNP frequency, so the absolute frequency values we report herein should be viewed with some caution. Nevertheless, the relative polymorphism frequency values reported herein support the hypothesis that long non-coding transcripts are less conserved than long protein-coding transcripts, perhaps as a function of selective constraints. We think that our modeling approach (The authors declare no conflict of interest.SHE and JAD conceived and designed the experiments. SHE performed the experiments and analyzed the data. JAD contributed reagents/materials/analysis tools. SHE and JAD wrote the manuscript. All authors read and approved the final manuscript.Contigs and SNPs. Information on 4,552 contigs with ≥10 sequence reads and ≥100 bp in length, and 2,980 SNPs found in the contigs.Click here for fileSupporting figures and tables. A file containing additional data: Figure S1; Figure S2; Table S1; and Table S2.Click here for file


New sequencing and genotyping technologies have enabled researchers to generate high density SNP genotype data for mapping populations. In polyploid species, SNP data usually contain a new type of information, the allele dosage, which is not used by current methodologies for linkage analysis and QTL mapping. Here we extend existing methodology to use dosage data on SNPs in an autotetraploid mapping population. The SNP dosages are inferred from allele intensity ratios using normal mixture models. The steps of the linkage analysis  are extended to use the dosage information. For QTL analysis, the probability of each possible offspring genotype is inferred at a grid of locations along the chromosome from the ordered parental genotypes and phases and the offspring dosages. A normal mixture model is then used to relate trait values to the offspring genotypes and to identify the most likely locations for QTLs. These methods are applied to analyse a tetraploid potato mapping population of parents and 190 offspring, genotyped using an Infinium 8300 Potato SNP Array. Linkage maps for each of the 12 chromosomes are constructed. The allele intensity ratios are mapped as quantitative traits to check that their position and phase agrees with that of the corresponding SNP. This analysis confirms most SNP positions, and eliminates some problem SNPs to give high-density maps for each chromosome, with between 74 and 152 SNPs mapped and between 100 and 300 further SNPs allocated to approximate bins. Low numbers of double reduction products were detected. Overall 3839 of the 5378 polymorphic SNPs can be assigned putative genetic locations. This methodology can be applied to construct high-density linkage maps in any autotetraploid species, and could also be extended to higher autopolyploids. Dactylis glomerataPaspalum notatum Flügge Linkage analysis and QTL mapping studies are now widely used in diploid plant species and are becoming increasingly common in polyploid species. Recent linkage maps for autotetraploid species include maps for potato New genotyping technologies, for example the Illumina Infinium platform employed here, as well as sequence based methods, such as RAD sequencing, or genotyping by sequencing (GBS) allow the generation of high density single nucleotide polymorphism (SNP) genotype data on mapping populations and other similar populations 1 offspring to use SNP dosage information generated using Illumina Infinium technology. A mixture of normal distributions is used to infer SNP dosage from the intensity data. Our approach differs from that of In this paper we extend the method of These methods are applied to a full-sib potato population derived from a cross between processing clone 12610ab1 and the cultivar Stirling. Previous maps of this population using AFLP and SSR data have been published by The population studied was the progeny of a cross between the breeding clone 12601ab1 and the cultivar Stirling. This tetraploid population has been extensively studied SNP genotyping was carried out on the parental clones , as well as 190 offspring using the Infinium 8303 potato SNP array DNA samples were arranged in two 96-well plates, each in an arrangement of 12 columns by 8 rows. The parents Stirling and 12601ab1 were in rows 1 and 2 of column 1 on plate 1, followed by the offspring. The intensities of the fluorescent dyes associated with the two alleles of the SNP are expressed as Cartesian coordinates  by the programme Genome Studio. After normalisation, Genome Studio transforms the intensities to a combined SNP intensity R = (X+Y) and an intensity ratio theta = (2/π)*arctan(Y/X) Three criteria were used to identify SNP theta scores as suitable for further analysis: a trimmed range greater than 0.1, no strong spatial trend and no missing values.Range: An ideal SNP for which all possible dosages  can be observed is expected to consist of theta scores in five clusters, centred around 0.0, 0.25, 0.5, 0.75 and 1.0 respectively. This has a range of 1.0. The lowest expected range is for a simplex SNP, for example with parents AAAA x AAAB, where parents and offspring will have scores clustered around 0.0 and 0.25, and so a range of 0.25. Inspection of the data suggested the use of a trimmed range, between the 2% and 98% quantiles, to remove the influence of a small number of outliers. SNPs were retained in the set for analysis if the trimmed range was greater than or equal to 0.1.Spatial trend: Index plots of the theta scores against the order of the samples on the plates showed that for some SNPs there was a spatial trend and/or a difference between the plates. To address this, the theta score was modelled as a smooth function of the sample order using locally-weighted regression (loess) with four degrees of freedom Missing values: There were a small number of SNPs with missing theta scores and for computational ease these were excluded.A normal mixture model can be used to model the theta scores of the offspring. This will have from one to five component distributions, depending on how many dosage categories are possible for that configuration. The mixture proportions will be considered as given, using the information in P1y and P2y for the parents and 1…yny for the n offspring. Let A be the more frequent allele, and B the less frequent. Let the mixture proportions for configuration C be C(g)p, where g = 0…4 corresponding to the possible dosages, expressed as the number of ‘B’ alleles. The likelihood of the offspring data under configuration C can be written asgμ and variance 2σ.Let the theta scores be The following derivation is adapted from The maximum likelihood equations areθ, so the first term is zero. The second term represents a regression of the offspring theta scores as a function of the dosage, and weighted by the probability of each dosage given the theta score C with k dosage classes, the algorithm consists of the following steps:For configuration k-mean cluster analysis C. For example, if configuration C is an AAAA x AAAB model, the initial cluster means are 0.0 and 0.25. The final cluster means are used for the next step.A Initial expectation step. Calculate the probability of each observed theta value arising from each dosage class Initial maximisation step. Carry out a weighted regression of the theta values on the dosages to obtain updated estimates of the dosage class means and variance.Repeat steps 2 and 3 until the log-likelihood converges.Allocate each offspring to a dosage class based on the maximum posterior probability C. The minimum Bayesian information criterion (BIC) C, including the null configuration, to determine which is the most likely. This is defined asp is the number of estimated parameters, equal to the number of class means plus one (for the variance).The full likelihood for the parents and offspring is then computed asA chi-square goodness-of-fit test is used to test the segregation of each SNP by comparing the numbers of offspring in each dosage class of the selected configuration with the theoretical proportions in While it is possible to cluster all the non-distorted SNPs together and then carry out a global linkage analysis of each linkage group, a preliminary step of ordering the simplex SNPs and making tentative associations of these into chromosomal groups for each parent is a useful check. This makes the later analyses, especially the overall cluster analysis, easier to interpret. Simplex SNPs can be scored as 1 or 0, depending on whether they carry the simplex allele or not. Recombination fractions between pairs of simplex SNPs are calculated in the same way as other simplex markers, using the equations given by r. In each case six genotype classes are found in the offspring. Work on dominant markers SNPs with significant distortion were excluded before clustering into linkage groups. For simplex SNPs, the threshold for exclusion was if the significance of the chi-square goodness-of-fit statistic was less than 0.001. Our experience with analysing other types of markers in tetraploid populations is that distortion in other marker configurations can give more problems in map assembly, and so for duplex and higher dosages the threshold applied was to exclude the SNP if the significance of the chi-square goodness-of-fit statistic was less than 0.01. The SNPs were clustered into linkage groups using a chi-square test for independent segregation as described by When pairs of SNPs are being considered, with different configurations and different phases, there are a very large number of possibilities. The expected class dosage frequencies could be derived for each configuration, as done for simplex+duplex pairs etc in r between them and write the parental genotype as SaTa/SbTb/ScTc/SdTd to represent the alleles on the four homologous chromosomes. At gamete formation, we assume that all three possible pairings into bivalents are equally likely, with probability 1/3: . Within each bivalent, there are three types of gamete: non-recombinant (four genotypes), single recombinant (eight genotypes) or double recombinant (four genotypes), with probabilities (1−r)2/4, r(1−r)/4 or r2/4. For the first bivalent pairing above, examples would be SaTaScTc, SaTaScTd and SaTbScTd respectively. A general equation for the frequency of a gametic genotype can be written asi0x, i1x and i2x are the numbers of nonrecombinants, single recombinants and double recombinants for genotype i.Let S and T be two linked SNPs with recombination fraction i0x, i1x and i2x are 4,8,4 for this gamete. The other two bivalent pairings gives heterozygous bivalents AC/BD and AC/BD, with nine possible gametes. The coefficients i0x, i1x and i2x are summarised in As an example, consider a parent with two duplex SNPs linked in coupling. Let SNP S have alleles A and B, and SNP T have alleles C and D, and let the dosage measure the number of B and D alleles respectively. The parental genotype is AC/AC/BD/BD. One possible bivalent pairing gives the homozygous bivalents AC/AC and BD/BD and the gamete ABCD is obtained regardless of the degree of recombination i.e. the counts i can be written asijy} for dominant or codominant markers was modified to give the coefficients associated with each dosage class.A random union of the gametes from the two parents gives zygotes derived from zero to four recombinations, and a general equation for the zygote genotype in offspring have genotype i, for genotype classes i = 1…k, the log-likelihood for the recombination fraction r can be written asr can be obtained by setting the derivative of the log-likelihood with respect to r equal to zero. Luo et al. r = r = 0.5 for the most likely phase.If Luo et al. The above analysis was used to estimate the most likely phase, the recombination fraction and the LOD score for all pairs of SNPs within each of the twelve linkage groups. This information can be used to order the SNPs and estimate the distances between them, for example using the weighted least squares procedure of The final step in the map construction was to reconstruct the phase of the complete linkage group. The simplex SNPs provide a framework for the linkage group, ideally identifying all four homologous chromosomes from each parent, and other SNPs were placed relative to these. A computer routine was written to check the most likely phase of each non-simplex SNP with respect to each simplex marker, and alleles were designated as in coupling or repulsion with each if the LOD score was greater than 5.0. Any SNP with an unknown or incomplete phase after this step had its phase assigned manually, using SNP pairs with as high a LOD as possible and also a high LOD difference between the most likely and the second most likely phase.The linkage map can be checked by QTL interval mapping of the original theta scores on the linkage map. This gives a check of the position of the SNP, and whether the phase has been identified correctly. For example, if a SNP has been identified as a simplex SNP on chromosome d of parent 1, we expect the theta score for that SNP to map to the SNP position, and for the coefficient associated with chromosome d of parent 1 to be the only significant coefficient.The approach used for QTL interval mapping is similar to that proposed by In order to carry out a QTL analysis, we need to estimate QTL genotypes in the offspring from the parental genotypes and phases, and the offspring dosages. There are 36 possible genotypes (assuming no double reduction). Previous work In this example, only one genotype is possible for each of SNPs M1 and M4, cdgh for M1 and cdfh for M4. For M2 there are 18 possible genotypes (i.e. every configuration including chromosome f). For M3, there are 15 possible genotypes that give the observed dosage. We infer that there must have been a recombination between M1 and M2, as M1 has chromosomes g and h from parent 2 and M2 has chromosome f. This is compatible with a bivalent pairing of f with g and e with h in parent 2. The genotype cdfh will give the observed dosages for SNPs M2 and M3 as well as M4. Therefore this sequence of dosages can be obtained from one recombination in parent 2, between SNPs M1 and M2, and no recombination in parent 1 over this section. All other possible reconstructions would involve more recombination events.K SNPs, we obtain a 36 by K matrix for each offspring indicating whether the observed offspring dosage can be obtained from that genotype or not.In general, for a chromosome with Hackett et al. A HMM has a series of states, forming a path. Transition probabilities model the transitions from one state to another. From each state, symbols are emitted, modelled by emission probabilities. Only the symbols are observable, and the aim is to infer the path of states from the observed symbols.i, i = 1…K} and the sequence of observed symbols as {ix, i = 1…K}. The transition probability between states is given byi.Denote the path of states as {πK along the chromosome, where positions here replaced the usual HMM ‘times’. Associated with each genotype is an indicator iks, equal to 1 if the offspring dosage at position i can be obtained from genotype k, and 0 otherwise. The observed symbols are a sequence of K symbols ‘y’, indicating correct matches between the genotypes and the offspring dosages. The set of emitted symbols from each state is {y,n}. If the SNP data is regarded as error-free, the emission probabilities areTo apply this to a single offspring, consider the 36 genotypes to be the unobservable states at SNP positions 1 to For example, consider bivalent pairing chromosome a with b and c with d in parent 1 and e with f and g with h in parent 2.i is aceg, then if there are no recombinations the process will remain in state aceg at position i+1. To move to state bceg, adeg, acfg or aceh requires 1 recombination. Other possible moves and the number of recombinations are shown in If the state at position r is the recombination fraction between positions i−1 and i, and kld is the number of recombinations between those states. A preliminary analysis used an approximate value of 0.01 for each recombination fraction, as there are approximately 100 SNPs mapped by JoinMap on each chromosome, with a length of approximately 100 cM. This is a regular HMM. The model was then extended to a position-dependent HMM, using the estimated recombination fractions for adjacent SNPs, and this is presented here.The transition probabilities are thereforeB of each of the different genotype states along the path,p(x|B), each conditional on a particular bivalent pairing. The overall genotype probabilities at each map position are then calculated asFrom the HMM, we need to estimate the probabilities under each bivalent model iAjAkAl as the sum of a population mean, the main effects of the alleles and diallelic, triallelic and tetraallelic interactions. For our full sib family model, with parent 1 having chromosomes a–d and parent 2 having chromosomes e–h, an individual will inherit alleles Ai and Aj from parent 1 with i,j taking values from . Likewise it will inherit Ak and Al from parent 2, with k,l taking values from . Let iX, i = a…h be 0/1 indicator variables corresponding to allele Ai being absent/present for that individual. As each individual inherits 2 alleles from each parent, we have the constraint that aX+bX+cX+dX = 2 and eX+fX+gX+hX = 2, and corresponding constraints on interactions among There are various models with different levels of complexity that can be used for quantitative traits. Kempthorne a…Ah be a….αhα. A main effects model, taking into account the constraints on iX, has the formIn modelling the theta scores for each SNP, we expect them to be a simple function of the number of B alleles at the corresponding SNP, and therefore we model them using the main effects of each allele in Kempthorne's model, assuming interaction terms to be zero. Let the main effects of alleles AA small simulation study was used to test how well the QTL genotypes were reconstructed by the HMM. Using the maps for each of the 12 chromosomes, 100 individuals were simulated so that their genotypes at each SNP were known. The HMM was used to estimate the genotype probabilities, and this was compared with the known genotypes.The theta scores for the SNPs used by JoinMap were then analysed, mapping each set of theta scores on the expected chromosome.For 6176 of the 8300 SNPs, the trimmed range for this population was greater than or equal to 0.1. Exclusion of SNPs showing a significant trend (with p<0.0001) reduced the set further to 5609 SNPs. There were only 23 SNPs among the remaining set with missing values. While most had fewer than ten missing values, for computational ease these were excluded. This gave a final dataset of 5586 SNPs.p<0.001. The proportion of distorted SNPs is higher for configurations with larger numbers of classes, suggesting that these may be harder to fit. A total of 5332 SNPs were classified as segregating, rather than the null configuration, and 520 of these were distorted with p<0.001. The theta scores for these 520 distorted SNPs were checked graphically and the BIC scores re-examined to see if these suggested that an alternative configuration had a similar BIC. This showed 46 SNPs that resembled simplex SNPs graphically (i.e. two similar sized groups), but where a small number of outliers or some trend led to the EM algorithm fitting more than two groups. These 46 were recoded as simplex SNPs to give a total of 5378 SNPs for linkage analysis.Normal mixture models were fitted to the theta scores for each of the 5586 SNPs obtained after pre-processing. For most SNPs one model was clearly indicated. Unevenly spaced mixture means were detected, but did not complicate the genotype designations. In general the posterior probabilities for allocating offspring to genotype classes were high, with a mean posterior probability of 0.982, and a mean number of 180.0 of the 190 offspring having a posterior probability greater than or equal to 0.90. p<0.001, and the addition of the recoded simplex SNPs, there were 1040 simplex SNPs from Stirling and 887 from 12601ab1. The SNPs from Stirling segregated into 51 groups of 4–65 SNPs, while the SNPs from 12601ab1 segregated into 45 groups of 4–75 SNPs. In general the groups were easily ordered, with few SNPs having problems with poor fit or many double recombinants. The recoded simplex SNPs were found to fit well on the maps.After exclusion of the SNPs that were distorted with p<0.001) were used to tentatively identify homologous chromosomes. For Stirling, eight groups with four homologous chromosomes, three groups with three homologous chromosomes, one group with five homologous chromosomes (two of which were very short) and one isolated group were identified. For 12601ab1, six groups with four homologous chromosomes, four with three chromosomes, two with five chromosomes  and one isolated chromosome were identified, giving twelve groups and one isolated chromosome for each parent. Significant simplex to double-simplex coupling pairs identified matches between the chromosomes from each parent, giving 12 tentative linkage groups.Duplex-simplex SNP pairs with significant non-independent segregations  markers in coupling phase have the highest LOD scores. For this study, these had a mean LOD score of 32.1 (41.3). Four configurations had higher mean LOD scores: XSS+XSS in coupling phase (mean LOD 43.1 (62.1)), SS+SS in coupling phase (mean LOD 40.5 (57.2)), DS+DS in coupling phase (mean LOD 38.9 (59.7)) and DD+DD in coupling phase (mean LOD 37.8 (58.6)). For the SS+SS configuration in coupling phase, this LOD can be compared with the expected LOD of 22.9 (32.6) for this population if the marker was scored as a dominant marker without dosage information . Sixteen SNP configurations had lower mean LOD scores than this. Most of these were configurations containing simplex SNPs in repulsion phase or duplex SNPs in mixed phase, for example a DD SNP linked to a NxD SNP in mixed phase (mean LOD 0.9 (1.1)), or a DS SNP linked to a DS SNP in mixed phase in the first parent and repulsion in the second parent (mean LOD 1.6 (2.6)). Other configurations with low mean LOD scores were a DS SNP linked to a SD SNP in coupling in one parent and repulsion in the other (mean LOD 2.3 (3.1)), and a DD SNP linked to a SS SNP in coupling in one phase and repulsion in the other (mean LOD 1.6 (1.8)). If a pair of SNPs with one of these less informative configurations were tested in isolation, they would be wrongly classed as unlinked. However in a larger set of SNPs with a range of configurations, linkage can be established via a third SNP in a more informative configuration.The configurations were also examined to see whether the correct phase was always inferred. For some SNP pairs, two phases are possible, but give the same recombination fraction and LOD score. An example is a pair of DD SNPs, linked in coupling in one parent and in mixed phase in the other, where the CxM and MxC phases cannot be distinguished unless information from additional SNPs is included. Apart from such pairs, the phase was inferred correctly for the configurations with high LOD scores. For the configuration DS linked to DS in CxR phase, with mean LOD 8.5 (11.6), the phase was inferred wrongly as MxC for one of the 20 simulated populations with recombination fraction 0.1 . However the recombination fractions and LOD scores were very similar. For the configuration SD linked to SS in CxR phase, with mean LOD 5.7 (8.6), the phase was inferred wrongly as RxC for one of the 20 simulated populations with recombination fraction 0.1, and for XSS linked to DS in CxC phase, mean LOD 5.5 (9.8), two of the 20 populations with recombination fraction 0.1 were wrongly inferred as RxR. For the simulations with true recombination fraction 0.05, the configuration SS linked to SD in RxC phase, with mean LOD 2.4 (3.7), the phase was inferred incorrectly for three of the simulated sets, twice as CxC and once as CxR. These were the only incorrect inferences of phase for configurations with a mean LOD score greater than 3, but incorrect inferences became more common for configurations with mean LOD score below 3.On the basis of this small study, we conclude that this approach identifies the correct phase and estimates recombination fractions precisely for all but the least informative configurations. Using dosage information leads to higher LOD scores than when the corresponding configuration is scored as presence/absence. For the least informative configurations, some linkages will be missed, and some phases may be inferred incorrectly, and the probability of this happening increases with increased marker separation, as for presence/absence markers. However incorrect inferences can be avoided by using information from additional markers in more informative configurations. We found no cases of an incorrect inference with a high LOD score.After removal of the near-duplicate SNPs, recombination fractions and LOD scores were calculated between all pairs of SNPs in each chromosomal group for each possible phase, and the recombination fractions and LOD scores with the highest log-likelihood were written to a pairwise data file for analysis with JoinMap 4. The following JoinMap options were used: the Haldane mapping function; using only pairs of SNPs with recombination fractions less than 0.45 and LOD scores greater than 0.05, and to run a ripple search after every five SNPs were added to the map. Two mapping rounds were run with JoinMap, and the JoinMap diagnostics to check the fit of the mapped SNPs were examined.It was found that XSS SNPs gave particular problems in the map assembly. As there were only a small number of these, it was decided to omit this class of SNPs from the JoinMap analysis. Isolated simplex SNPs placed into a chromosomal group by the cluster analysis but unlinked to any of the other simplex SNPs were also omitted from the JoinMap analysis. The map obtained after two rounds of JoinMap was used as the first estimate of the linkage map (the ‘mapped’ SNPs). Other SNPs were placed in approximate bins according to their maximum LOD with the mapped SNPs, provided the recombination fraction was less than 0.05. These are referred to as the binned SNPs.SNP phases were estimated for the mapped and binned SNPs as described above. The QTL mapping approach (see details below) was then used to map the theta scores for the mapped and binned SNPs as quantitative traits on the linkage map of mapped SNPs. Where discrepancies were identified, the SNPs in question were removed and the JoinMap analysis and QTL analysis were rerun. K, where K is the number of SNPs on the chromosome). It also summarises the proportion of genotypes where three chromosomes out of four were correctly identified. The proportion reconstructed correctly was greater than 0.75 for nine of the twelve chromosomes. The three chromosomes with lower proportions reconstructed correctly had the highest proportions of simplex SNPs, which are only informative about the presence/absence of a single chromosome. They also were among the least dense of the maps. Although it was not investigated here, it would be possible to include additional higher dosage SNPs from the binned set to improve the coverage for these chromosomes.A small simulation study was used to check how well the HMM reconstructed individuals with known genotypes, simulated from the final SNP maps. One hundred offspring were simulated for each chromosome. QTL mapping was then used to check the position, the %variance explained and the inferred phase of the mapped and binned SNPs when their theta scores were mapped as quantitative traits. We will look at chromosome V in detail. This had 396 SNPs according to the cluster analysis, of which 158 were removed as near-duplicates, giving 238 for the JoinMap analysis. Eleven XSS SNPs and eight isolated simplex SNPs were also excluded before the JoinMap ordering was run. The first analysis mapped 99 SNPs, and left 120 unmapped. Six of the unmapped SNPs were too far from any of the mapped SNPs to be placed in bins, but the remaining 114 were allocated to bins. All the mapped SNPs, binned SNPs and the 19 SNPs excluded from the JoinMap run were analysed by QTL mapping of the theta scores.For the mapped SNPs, the %variance explained ranged from 85.6% to 99.4%, with a mean of 94.0%. One SNP mapped as a QTL to a position 11 cM from its location as a marker, but the other 98 mapped SNPs had less than 5 cM between the locations as markers and as QTLs, with mean displacement 0.8 cM. For the binned markers, the %variance explained ranged from 77.0% to 99.4%, with a mean of 92.3%. There were greater displacements between the locations as markers and as QTLs, with a mean displacement of 3.9 cM and a maximum displacement of 31 cM. Two of the SNPs excluded from the JoinMap analysis had lower %variance explained, at 65% and 68% respectively. For comparison, 100 random traits generated from a standard normal distribution were analysed by the same approach to look at the distribution of the %variance: this had a mean of 4.0% and an upper 95% point of 9.4%.All SNPs with %variance <85% or a discrepancy between the SNP phase and the significant QTL coefficients were inspected graphically. A further 33 SNPs  were excluded as a result of this process, due to trend or excess scatter leading to uncertainty in the SNP genotype inference. The JoinMap ordering was rerun without these SNPs, giving a final map of 119 mapped SNPs, 65 binned SNPs and two unmapped SNPs that were too far from the mapped SNPs to be placed in a bin. The QTL mapping analysis was rerun and no further problems were found. The mean displacement between positions as markers and positions as QTLs of the remaining binned markers decreased to 3.0 cM, with a maximum displacement of 14 cM. This is still higher than for the mapped markers, probably due to the lower precision of the position of the binned markers. d between the theta scores for the two dosage classes. A double reduction product is expected to be separate from these classes i.e. to have a large residual. Any SNP where there were residuals greater than 0.75*d were inspected graphically and the identities of individuals with double reduction products noted. An example is given in This analysis has assumed a model of random chromosomal segregation, with bivalents forming and recombination within these. However segregation in autotetraploids is more complicated, as reviewed by This analysis was complicated by trend in the theta scores for some SNPs. It was not always clear whether extreme values were due to double reduction products or to trend. Double reduction was therefore inferred only for individuals with a double reduction product for two or more SNPs linked in coupling on the same homologous chromosome but at separate locations. For Stirling, double reduction products were detected for all but one chromosomal group. The maximum number of individuals where double reduction has occurred on a chromosomal group (i.e. on any of the 4 homologous chromosomes) was six of the population of 190, with a mean number of 2.7 individuals. For 12601ab1, double reduction products were detected for eight of the twelve chromosomal groups. The maximum number of individuals was 13, with a mean of 3.4 individuals. We conclude that the rate of double reduction in this population is low, but could be usefully explored more once issues of trend have been addressed.In this paper we have developed statistical methodology to construct SNP linkage maps in autotetraploid species, making use of the additional information given by the SNP allele dosages, in contrast to current methods based on allele presence/absence data. This permits the use of SNP configurations that contain little or no information with presence/absence data. For example, a triplex SNP AAAA x ABBB will result in offspring that all carry both the A and B alleles (unless double reduction occurs), but dosage information will separate the AAAB offspring from the AABB, so that this SNP can be mapped. Similarly, for a simplex by duplex SNP AAAB x AABB, all offspring will carry the A allele and the ratio presence to absence of B will be 11∶1 (in the absence of double reduction), which is too high to be informative, while the dosage data identifies four classes with 0–3 B alleles. Another advantage is that the precision of the estimates of recombination fraction may be increased by the use of dosage information compared to presence/absence information: we have demonstrated this for the case of duplex and double-simplex SNPs. The methods developed here are suitable for linkage mapping of any genotypic data derived from a technology that is informative about allele dosages for tetraploid species. We anticipate that these methods can be further extended to look at dosages derived from read counts obtained from genotyping by sequencing The methods have been applied to construct a linkage map in the well-studied potato cross between cultivar Stirling and breeding line 12601ab1, using 190 progeny. The analysis was complicated by the discovery of some spatial trend related to sample order in the allele intensity ratios (theta scores) derived from the SNP technology employed (Illumina iSelect) as well as the Genome Studio software. This was most apparent in the non-segregating SNPs with a low range for the theta scores, but could also be found in some segregating SNPs. SNPs with a strong trend were excluded from the analysis, but some further SNPs with trends were detected and removed later, when theta values were mapped as quantitative traits. All of this analysis has been carried out using with summary-level data from Genome Studio, i.e. one observation per probe type, per sample genotype. Ritchie et al. Methodology has also been tested for QTL mapping, using a hidden Markov model to infer QTL genotype probabilities in the offspring and modelling the traits as a function of these. So far these methods have only been applied to theta scores, where it is expected that very high proportions of the variance should be explained by a single location on the linkage map, and that the effects of the separate alleles on the traits are likely to be additive. Further testing is in progress to establish suitable thresholds for using this method for regular phenotypic traits, controlled by several QTLs and with a higher proportion of environmental variation, and to compare different models for the traits, including interactions between alleles.The map constructed here contains 3839 mapped markers, either mapped directly or in ‘bins’. Previous maps constructed for this population contained far fewer markers (<500), which were primarily dominant AFLPs, with a smaller number of co-dominant SSRs These data can be made available to other researchers: please contact the authors for details.Figure S1Comparison of the tetraploid linkage map for chromosome I with that estimated on diploid potato population DRH by Felcher et al. .(TIF)Click here for additional data file.Table S1Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S2Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S3Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S4Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S5Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLSX)Click here for additional data file.Table S6Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S7Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S8Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S9Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S10Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S11Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.Table S12Order of the mapped, binned and duplicate SNPs on chromsomes I–XII. The (near) duplicate and binned markers are listed below the closest mapped marker. The recombination fraction of each binned marker with the closest mapped marker is shown in the final column.(XLS)Click here for additional data file.