Phased genome maps are important to understand genetic and epigenetic regulation and disease mechanisms, particularly parental imprinting defects. Phasing is also critical to assess the functional consequences of genetic variants, and to allow precise definition of haplotype blocks which is useful to understand gene-flow and genotype-phenotype association at the population level. Transmission phasing by analysis of a family quartet allows the phasing of 95% of all variants as the uniformly heterozygous positions cannot be phased. Here, we report a phasing method based on a combination of transmission analysis, physical phasing by pair-end sequencing of libraries of staggered sizes and population-based analysis. Sequencing of a healthy Caucasians quartet at 120x coverage and combination of physical and transmission phasing yielded the phased genotypes of about 99.8% of the SNPs, indels and structural variants present in the quartet, a phasing rate significantly higher than what can be achieved using any single phasing method. A false positive SNP error rate below 10*E-7 per genome and per base was obtained using a combination of filters. We provide a complete list of SNPs, indels and structural variants, an analysis of haplotype block sizes, and an analysis of the false positive and negative variant calling error rates. Improved genome phasing and family sequencing will increase the power of genome-wide sequencing as a clinical diagnosis tool and has myriad basic science applications. Construction of completely phased genomes remains difficult to achieve. Statistical approaches based on the exploitation of the haplotype structure of human populations Both physical and genetic transmission methods can be used to formally determine the phasing of variants. Physical methods are based either on experimentally separating the two haploid genomes prior to sequencing, or on pair-end sequencing of library of DNA fragments. Phasing has been inferred by imaging single molecules after long-range PCR Genome-wide transmission analysis to infer phasing was pioneered by Roach et al. on a quartet cis-acting regulatory sequence instead. Identifying the target genes of such variants is difficult. Phased genomes can help identify functionally significant variants in cis-acting regulatory sequences particularly if allele specific functional data such as RNA-Seq are also available. Phasing information helps determine whether double heterozygous mutations affecting the same transcription unit are in cis or trans. This is critical since in the former case only one of the two alleles is likely to be functionally deficient, while in the latter case, both alleles could be deficient depending on the nature of the effect. Finally, phased data is also invaluable to understand gene flow and characterize variant effect at the population level.Deciphering the phase of all variants after whole genome sequencing is important for a number of reasons. First, several genetic diseases are dependent on the parental origin of alleles. Phased genomes greatly facilitate the identification of the genetic variants responsible for such diseases. Additionally, SNPs and structural variants causing disease are not all located within a protein coding sequence, but can be within Using current algorithms published by Roach et al and Dewey et al., state of inheritance analysis in quartet allows the phasing of about 95% of all variants in Caucasians because variants that are uniformly heterozygous in the four family members cannot be phased in any of the family members. In more ethnically diverse families the proportion of variants that can be phased by transmission is lower because the proportion of heterozygous variants is higher. Moreover, transmission analysis does not allow complete phasing of the parents chromosomes: Rather, the parent chromosomes can only be phased within blocks defined by the location of the crossing-overs detected in the children .Here, we report a genome phasing method based on combining transmission analysis with analysis of pair-end reads from fragment libraries of different sizes and with population-based analysis. Combination of these three methods allowed us to phase almost all variants including 99.89% of the heterozygous SNPs and 94.4% of the quadruple heterozygous. We provide a complete analysis of the number of SNPs, indels and structural variants in the quartet, a detailed characterization of the rates of false negative and false positive calls and an analysis of haplotype sizes. We also provide an analysis of genes homozygously affected by SNPs, indels or structural variants expected to severely disrupt protein sequences.9 100-mer reads were obtained . About 70% of the uniquely aligned bases were reported with a quality score above 35. The alignment rate averaged 90% for the pair-end libraries but was lower (about 48%) for the mate-pair libraries because of the complex structure of these libraries. SNPs and indels were called using the GATK software package Informed consent from a healthy Caucasian family quartet  we classified all SCEs and MIEs as sequencing errors. We did not attempt to detect de novo mutations.Once inheritance state blocks were generated, we performed an analysis of transmission errors to calculate the number of true SNPs in the quartet and to compute error rates. Two major types of errors can be detected by transmission analysis: Mendelian Inheritance Errors (MIE) and State Consistency Errors (SCE). MIEs are sequencing errors that lead to a combination of genotypes within the quartet that are incompatible with Mendelian inheritance laws. For instance, an MIE will be detected when the parents are both homozygous and identical to the reference sequence and at least one of the two children is heterozygous for a SNP . SCEs arTransmission error analysis can be used to estimate the total number of SNPs and indels in a quartet and the rate of false negative calls, if the proportion of errors that are genetically detectable is known. As discussed in supplementary methods  and in aUsing the GRCh37 (hg19) human genome assembly and the GATK Unified Genotyper module on high-quality reads and without additional filters, we detected 5,163,231 fully-called genomic positions where a SNP was present in at least one of the four family members . In addiTransmission error analysis revealed 381,418 GDEs  MIEs and 227,866 (4.32%) SCEs). Since the number of GDEs represents 75% of all errors, it follows that there are at least 4,654,674 SNPs in the quartet  minus 381,418 (the number of GDE) and minus 127,139 ). Subtracting the GDEs from the list of called SNPs provided a list of 4,781,813 GDE-corrected SNPs. This list of SNPs is referred to in the remaining of the text and in the figures as the 4.8 million SNP list.From this list it is possible to compute a rate of false positive calls defined as positions where at least one of the genotype in a family member is wrong: Since there are 127,139 undetected errors in the 4.8 million SNP list, the rate of false positive calls is equal to 2.66% . To decrease the number of false positive calls we successively applied a series of filters. We first applied a read-depth filter created using GenPlay We then applied the variant quality recalibration filter from the GATK which uses Gaussian mixture modeling of high-quality datasets to evaluate the probability that each call is real. This filter had an efficiency of 0.5 and decreased the false positive rate to 0.48% while increasing the false negative rate to 13.64%.Finally, to decrease the number of false positive calls even further, we applied a filter based on the GATK-generated PL values. These values are measures of the likelihood that different genotypes, from the one reported, are correct. Using this filter we were able to decrease the false positive rate to 0.09% at the expense of a false negative rate of 23.17% (filter efficiency = 0.15). With this most stringent filter, 3,882,210 SNPs and approximately 3,575 undetected errors remained, leading to a false positive error rate per genome and per base equal to about 2.78*E-7. This list of SNPs is referred to in the remaining of the text and figures as the 3.9 million SNP list.In summary, we detected about 4.8 million SNPs in the quartet with a false positive rate of around 2.66%. Filtration of these errors was possible and yielded a list of about 3.9 million SNPs that had very few false positives, but a high rate of false negatives. Both filtered and unfiltered lists of SNPs are useful for different applications. The number of SNPs that we found in this Caucasian quartet is slightly higher than what was previously reported: Roach et al. found 3,665,772 SNPs including 84,958 GDEs when comparing their quartet to the GRCh37 reference sequence, and Dewey et al. reported about 4,302,405 SNPs between their north European quartet and GRCh37. Since false negative error rates were not reported in these previous studies, and since Caucasian origin is a very loosely defined term, these numbers of SNPs cannot be directly compared.When no filters were applied 95.55% of the SNPs were in DbSNP with a Ti/Tv ratio of 2.07 . The 4.4We then estimated the false positive error rate using three other methods . First wAn important question was whether our false negative rate could be confirmed by using an independent technology. To answer this, we genotyped the father  are analyzed. Hemizygous corrections were performed after calling the structural variants (see below) and resulted in the correction of 605 MIEs in the highly filtered list of SNPs .Physical phasing was performed using the read-backed-walker from the GATK . When thCombining physical and transmission phasing methods allowed us to phase, on average, an additional 44% of the fully heterozygous SNPs in each quartet member. This increased the overall phasing rates to 98% without using any statistical inference method . In addiAllelic chromosomal segments in one or more individuals can be identical by descent or by ancestry. Genomic segments that are identical by descent are DNA fragment that can be traced directly to known ancestors such as parents and grand-parents. Genomic segments that are identical (or very similar) by ancestry are segments that contain variants in strong linkage disequilibrium because they were inherited from a common ancestor that lived an undefined number of generations ago. Genomic segments highly similar because of ancestry are called haplotypes and have been mapped using statistical inference in the major human populations. Haplotypes can be used to phase variants in the absence of any family information or can be used to complement transmission analysis. Haplotypes are also very useful in population genetic to track gene flow and understand the evolution of the human genome.To phase the remaining unphased quadruple heterozygous variants and to impute the MIEs, the SCEs ad the uncalled positions, we then run Beagle Beagle analysis was then used to phase the recombination blocks in the parents that cannot be phased by transmission. Twenty-five junctions were phased with a high degree of confidence because the cross-over took place between regions carrying different haplotypes in the father. The remaining 29 junctions were phased with a lower degree of confidence because the two paternal chromosomes carried the same haplotypes in the region where the crossing-over occurred. In the case of the mother 26 junctions could be phased with a high degree of confidence. Once the blocks were phased a new tag, termed the AA tag, indicating the parental allele of origin for every SNPs, indels and SVs was added in the vcf file to the format field of almost all variants.As discussed above, haplotypes are important tools to track gene-flow and to understand genotype-phenotype relationships. When a phased genome is available, the size of each haplotype block, that is the size of the segment of DNA that are highly similar because of ancestry can be directly measured by comparing the phased chromosomes.To visualize the haplotype blocks, we plotted three parallel tracks in GenPlay the homozygous SNP (1|1 genotypes) and the SNPs that were present on either of the chromosomes (0|1 or 1|0 genotypes). As expected, this revealed a block structure that is equivalent to the haplotype structure  that canAs shown in Calling indels with the GATK yielded 714,192 indels including 123,289 GDEs  suggestiAccuracy of variant calls can be estimated by plotting histograms of read-depth for the different genotype calls. To compare the SNP and indels calls accuracy, we plotted the read-depth frequency histograms for the 0/0; 1/1 and 0/1 calls . In the de novo assembly analysis, we could not detect insertion of novel sequences in the genome. MIE analysis could not be performed to assess error rate since compatibility with Mendelian laws was used as a criteria to call the SVs. State of inheritance analysis revealed that about 8.1% of the CNVs in the list were SCEs. Examination of these variants revealed that the vast majority were not false positives, but were located in complex genomic regions that contained multiple deletions in different quartet members that had led to genotype miscalling. These regions were labeled as SCEs and were not studied any further.Once we had called the SNPs and indels using the GATK, we called the structural variants using a combination of Pindel We detected 1,459 deletions that were heterozygous in at least one of the four members of the quartet . As showWe also detected 72 duplications  includinWe also called 3,718 regions as deletions or duplications that were homozygous in the four members of the quartet . FurtherAnalysis using the SNPeffect cis or in trans. Out of an average of 404.25 double heterozygous high or moderate-impact mutations in each individual, 107.25 (27%) were in trans and 297 in cis (73%). The disproportionally high number of variants that were located in cis is caused by the haplotype structure of the human genome. This data demonstrate the value of phasing information for the evaluation of SNP effects.A question that can only be answered when phasing information is available is whether double-heterozygous mutations affecting the same gene are located in Although the number of indels is only about 6% of the number of SNPs, an average of 96.75±1.5 genes or isoforms was affected by indels classified as high-impact . This diAbout 10% (144/1571) of the deletions affected at least one protein coding genes and about 8% (107/1571) affected at least one exon. There were between 5 and 9 homozygous deletions that affected protein coding genes in each quartet member.Adding together the effects of the three variants type analyzed, there were about 200 genes homozygously affected by a high-impact variant in each member of this quartet of healthy individuals. We then analyzed the variant effect on each splicing isoforms. Computation of the number of genes that had all their isoforms affected by a high impact variant yielded a list of 41 genes  because We report here a method combining both genetic, physical and population-based approaches to phase the genomes of a quartet. The four haploid genome sequences that we have generated are among the most accurate published so far and false negative rates are reported for the first time. When transmission phasing was used alone, the phasing rate for heterozygous SNPs was about 91.5%. When genetic, physical and population-based methods were combined, the phasing rate reached 99.8%: In each individual, less than 4,800 SNPs were unphased out of average of about 1,800,000 million heterozygous SNPs.Haplotype blocks were defined using an island calling algorithm to quantify the variation in SNP density along the chromosomes. When more phased genomes become available, such data should help reconstruct the evolutionary history of each haplotype block and help interpret association studies.Using transmission error analysis, we were able to demonstrate that there are about 4.66 million true SNPs in the analyzed quartet. We provide a list of 3.9 million SNPs with virtually no false positive calls, but a large number of false negative calls (23%), and a list of 4.8 million SNPs with a higher false positive error rate but very few false negatives. Higher coverage or better sequencing technology would help decrease the false negative rate without increasing the false positive rate. Indel calling was significantly less accurate than SNP calling.Transmission error analysis was very useful to call and genotype the structural variants. We generated a list of more than 1,450 genotyped structural variants. Availability of the quartet sequence was critical because it provided internal read-depth control.Availability of phasing information allowed us to show that about 73% of all double heterozygous moderate or high impact SNPs that occurred within a transcription unit were compound heterozygous and therefore carried a wild type allele. Although all four members of the quartet are healthy individuals, we detected almost 200 high-impact variants within protein-coding genes in each member of the quartet.Is family sequencing worth the expense? Quartet sequencing allowed us to achieve a very high rate of phasing and to define our false positive and false negative error rates. Our analysis required the use of three phasing methods and the construction of libraries of different sizes. Family information is essential to eliminate errors and calculate error rates. Generally read-back phasing was relatively inefficient, at least with the library sizes and the relative coverage for each library that we obtained. Higher read-backed-phasing efficiency might be obtained by adjusting the library sizes and the number of libraries sequenced. Sequencing a quartet and omitting the read-backed phasing step would eliminate the need to generate multiple libraries and would yield similar error rate but would slightly decrease phasing accuracy since SNPs that are not in the HapMap (about 6% of the detected SNPs) are not phasable by this approach.It is difficult to estimate the cost of the production of a genome with similar precision by sequencing a single individual. Quartet sequencing at 120x coverage is roughly equivalent to sequencing two genomes at 60x coverage since the children genomes are contained within the parent genomes. In addition, the precision of the analysis is increased firstly because the GATK variant calling precision is higher when a quartet is sequenced since the variants are called as a group, and secondly because the availability of four genomes allows the generation of a highly efficient read-depth filter. Finally, at equivalent coverage, the number of undetectable errors would be about 4 times higher since transmission analysis allows the detection of about 75% of all errors. Phasing accuracy would also be much lower since as shown in In summary, when family members are available, quartet sequencing combined with population-based phasing is a powerful, cost-effective method to obtain genome sequences with low error rates and with precise phasing. Read-backed phasing allows a refinement of the phasing but requires additional library construction and additional analysis. Production of a genome sequence with equivalent precision from a single individual would cost much more than the cost of quartet sequencing.Genomic DNA was prepared from 10 ml of whole blood using Wizard Genomic DNA Purification Kit (Promega), quantified using NanoDrop and PicoGreen method .Genomic DNA was sheared to 300 bp and 600 bp size using Covaris, purified, and checked on bio-analyzer for size distribution. Each sample was end-repaired , an A-overhang added , and ligated to illumine pair-end adapter sequences (Illumina). The ligated libraries were size selected , PCR amplified by 10 cycles, and purified with SPRI beads . Libraries were quantified on a 1000 DNA Chip by 2100 bio-analyzer  and sequenced on Illumina HiSeq 2000.Genomic DNA was sheared to 2.5 to 3 kb in size using Covaris, purified, and checked on bio-analyzer for size distribution. The mate pair libraries were generated as indicated in the Illumina protocol . Briefly, Following DNA fragmentation, 2.5 kb fragments were end-repaired with biotin labeled dNTPs. The DNA fragments were circularized, and non-circularized DNA was removed by digestion. The Circular DNA was then fragmented to a size of about 300bp and biotin labels of the fragments  were affinity purified. Purified fragments were end-repaired and ligated to Illumina Paired-End sequencing adapters.Genomic DNA from one of the family members was hybridized to genome-wide human SNP array 6.0 (Affymetrix). Analysis was performed using Genotyping Console software (Affymetrix). The software uses birdseed algorithm and hg19 annotation files to generate the final SNP list.Alignments were performed using BWA and Novoalign. Default parameters were used for both aligners. BWA was used for all alignments except for the mate pair libraries which were aligned with Novoalign. All alignments were performed against GRC37 (hg19). Results of the alignments are described in SNP and indel calling was performed using the GATK software suite (version 1&2). Parameters were generally set according to the suggested best practice. Read quality (Q) was set to 35 during the SNP calling step. Once the variants were called they were sequentially filtered as follows:After alignment, a number of regions had abnormally high read-depth in all members of the quartet. These regions are mostly segmental duplications that were not incorporated in the reference genome. Variants cannot be genotyped in these regions because the aligned sequence represents the consensus sequence of several repeats that are not present in the reference genome. Attempt to genotype them yielded regions with high SCE density. To filter these error prone regions with high read-depth, we generated a filter that eliminates all the genomic regions that have higher than normal read-depth in all quartet members. . The filter was created using CNVnator and the Island finder algorithm of GenPlay. About 18.8 * 10E6 bases, that is 0.66% of the genome (assuming a genome size of 2.86*10E9 non N bases) is covered by this filter.Variant recalibration was performed following the best practice suggested by GATK.Rate of MIEs and SCEs was used as a guide to maximize specificity defined as the ratio of MIE to non-MIE variants eliminated. All regions that were uncalled in at least one family member were filtered out at this step . The most efficient filters were found to be filters that eliminated calls in which at least one family member had a PL value below a certain value. Result for PL >10, 20 or 50 are summarized in Hemizygosity creates MIEs that can be corrected once the deletions causing the hemizygosity are correctly genotyped. 605 MIEs caused by hemizygosity in some of the individuals were corrected.Variant effects were analyzed using a combination of SNPeffect . SVs were considered true positives if they were found by at least two programs in any of the group of files searched. Variants were considered true in the case which were found by two of the programs if they overlapped, regardless of the length. Overlap detection was performed using GenPlay and the Galaxy server. When variants overlapped they were given the sizes of the Pindel variants when available, or when Pindel had not detected the variants, the larger of the BreakDancer or CNVNator were used. Variants were extensively manually curated at the end of this analysis using read depth information and MIE information to eliminate incorrectly called or genotyped variants.Structural variants were detected using Pindel http://www.ncbi.nlm.nih.gov/bioproject/191008) with Accession number: PRJNA191008 and ID: 191008 or at http://genplay.einstein.yu.edu/FNY01/Click here for additional data file.Figure S2Triple heterozygous phasing. The preliminary inheritance blocks were used to phase the positions in each of the two trios that could not be phased without prior knowledge of the inheritance blocks. After this final phasing step, the edges of the preliminary blocks were refined.(TIF)Click here for additional data file.Figure S3Error analysis. (A) Mendelian Inheritance Error: quartet genotype is incompatible with Mendelian laws. (B) State Consistency Error: quartet genotype is compatible with Mendelian laws but pattern of chromosomal inheritance is not.(TIF)Click here for additional data file.Figure S4Physical phasing. Heterozygous SNPs found on both ends of overlapping pair-end or mate-pair library fragments can be phased. Read-backed phasing was performed using the GATK package.(TIF)Click here for additional data file.Figure S5Comparison of HapMap-based and quartet-based phasing. Chromosome 1 of individual FNY01_2_2 was phased using Beagle using the hapmap 3.0 marker file and default parameters and the results were compared with our quartet based phasing. A value of one was assigned to concordant phasing between both methods and a value of negative one for the discordant phasing calls. The switch accuracy defined as the proportion of heterozygous positions misassigned relative to the previous heterozygous position (TIF)Click here for additional data file.Figure S6Histograms. (A): Read-depth frequency histograms for 0|0 , 1|1  and heterozygous calls in individual FNY01_2_2 are plotted for SNPs and indels. The 4,8 million SNP list was used after application of the read depth filter. The histograms are almost perfectly super-imposable, suggesting that the alternate (ALT) calls do not occurs preferentially in regions of low read depth. Interestingly the allele depth for the MIEs and SCEs is lower suggesting that the SNP calling errors occur more frequently in regions of low coverage. Hence, higher coverage could decrease the error rate even further. In the case of the indels, the read-depth is in average higher for the reference (REF) calls than for the ALT calls. This confirms that indel calling by the GATK is less precise than SNP calling. (B): Allele-depth frequency histograms. Log ratios of allele-depth are plotted. ALT calls #/REF calls # are plotted in the case of 0|0 and heterozygous genotypes calls. REF calls #/ALT calls # are plotted in the cases of the 1|1 calls. Histograms were generated using either the 4.8 million list of SNPs after application of the read depth filter only, or on the 3.9 million SNP list . Table on the right summarize average allele-depth ratios. In the case of the SNPs, average allele-depth ratios are very small for the 0|0 and 1|1 calls and close to 1 in the case of the heterozygous calls. As expected, application of more stringent filters yields smaller allele depth ratio in the case of the homozygous calls, and a ratio that is closer to 1 (from 0.93 to 0.98) in the case of the heterozygous calls. We conclude that, as expected, filtering erroneous calls increase the allelic balance quality of the calls. Indel calling is of lower quality than SNP calling. These results also suggest that alignment with BWA followed by variant calling with the GATK yields results with minimal bias in favor of the reference allele.(TIF)Click here for additional data file.Figure S7Quality score. Average quality scores and average quality scores greater than 35 for each individual. Average of the results for three libraries (of three different sizes), sequenced once or twice each, and is shown for each individual. X-axis: quality score. Y-axis: number of reads.(TIF)Click here for additional data file.Table S1Alignment results.(XLSX)Click here for additional data file.Table S2Crossovers and Blocks.(XLSX)Click here for additional data file.Table S3Variant filtration. GATK calls were filtered to decrease the rate of false positives (see (XLSX)Click here for additional data file.Table S4Error Rate in the Identical Blocks.(XLSX)Click here for additional data file.Table S5Error Rate in the Non-Identical Blocks.(XLSX)Click here for additional data file.Table S6Error Rate in the Invariant Positions.(XLSX)Click here for additional data file.Table S7Affymetrix Validation. To validate the GATK calls and the error rate estimations we genotyped one of the individual of the quartet using the Affymetrix platform and calculated the discordance rate between the two platforms with or without filtration of the SNPs. The first row summarizes the results without filters: Of the 494,844 SNPs called by both platforms 493,014 were concordant and 1830 discordant (discordance rate = 0.37%). When the filters that reduce the false positive rate to 0.09% were applied (second row) the discordance rate drops even further to 0.21%.This is the expected result since the filters eliminates false positive calls. In the third row the SNPs that were filtered in the second row are analyzed. In that population of SNPs the discordance rate increased a little bit to 3.1% also as expected since the filtered SNPs should be enriched in erroneous calls. Importantly, the concordance rate in this list is above 96% suggesting that the vast majority of the filtered SNPs are false negative as predicted by the state of inheritance analysis. This result strongly validates our estimated of the rate of false negative and therefore of the total number of SNPs in the quartet. In the three last rows we analyzed the discordance rate for the SNPs that were flagged by the state of inheritance analysis as GDE. As expected the discordance rate is much higher (above 30%). The discordance rate is lower than would expected than by chance alone because calls were flagged as GDEs when an error was present in any one of the four quartet members. Therefore some of the calls flagged as MIE and SCE for the quartet might in fact be accurate in the particular individual (2_2) analyzed with the Affymetrix platform.(XLSX)Click here for additional data file.Table S8Read-Backed Phasing N10 and N50.(XLSX)Click here for additional data file.Table S9Genetic and Physical Phasing Comparison.(XLSX)Click here for additional data file.Table S10Parent Block Phasing.(XLSX)Click here for additional data file.Table S11List of deletions larger than 100 bp and genotypes in the four quartet members.(XLSX)Click here for additional data file.Table S12List of duplications larger than 100 bp and genotype in the four quartet members.(XLSX)Click here for additional data file.Table S13List of regions with abnormal read-depth that were genotyped as homozygous in the four quartet members. In addition to the deletions and duplications summarized in (XLSX)Click here for additional data file.Table S14Homozygous Variant Effects. List of the protein-coding genes for which all known isoforms are affected by a homozygous high-impact SNP, indel or structural variant in at least one quartet member.(XLSX)Click here for additional data file.Table S15Error Simulation for Sporadic Errors (Absolute Number of Genotypes). To estimate the number of errors that are detectable by transmission analysis, we simulated all possible genotypes in case of sporadic  or. Averaging the results of the simulation suggests that about 79% of all errors are detected in the children and 69% in the parents (see supplementary methods in (XLSX)Click here for additional data file.Table S16Error Simulation for Non Sporadic Errors. To estimate the number of errors that are detectable by transmission analysis, we simulated all possible genotypes in case of non-sporadic  and determined the number of genotypes that contained either an MIE or an SCE.(XLSX)Click here for additional data file.Text S1Supplementary methods and file description.(DOCX)Click here for additional data file.


New sources of genetic diversity must be incorporated into plant breeding programs if they are to continue increasing grain yield and quality, and tolerance to abiotic and biotic stresses. Germplasm collections provide a source of genetic and phenotypic diversity, but characterization of these resources is required to increase their utility for breeding programs. We used a barley SNP iSelect platform with 7,842 SNPs to genotype 2,417 barley accessions sampled from the USDA National Small Grains Collection of 33,176 accessions. Most of the accessions in this core collection are categorized as landraces or cultivars/breeding lines and were obtained from more than 100 countries. Both STRUCTURE and principal component analysis identified five major subpopulations within the core collection, mainly differentiated by geographical origin and spike row number (an inflorescence architecture trait). Different patterns of linkage disequilibrium (LD) were found across the barley genome and many regions of high LD contained traits involved in domestication and breeding selection. The genotype data were used to define ‘mini-core’ sets of accessions capturing the majority of the allelic diversity present in the core collection. These ‘mini-core’ sets can be used for evaluating traits that are difficult or expensive to score. Genome-wide association studies (GWAS) of ‘hull cover’, ‘spike row number’, and ‘heading date’ demonstrate the utility of the core collection for locating genetic factors determining important phenotypes. The GWAS results were referenced to a new barley consensus map containing 5,665 SNPs. Our results demonstrate that GWAS and high-density SNP genotyping are effective tools for plant breeders interested in accessing genetic diversity in large germplasm collections. Hordeum vulgare subsp. vulgare) was one of the first cereals that human communities of the Fertile Crescent domesticated about 10,000 years ago http://faostat.fao.org). Barley is essential for the malting and brewing industries and it is an important animal feed. It also constitutes a staple food in several regions of the world due to adaptation to high altitudes, drought and soil salinity Barley , one of the genes controlling spike architecture in barley Being one of the most widely adapted crops, the barley germplasm pool has the potential to contain enough genetic diversity to breed for adaptation to different environmental conditions. Moreover, the ample barley germplasm resources available worldwide www.ars-grin.gov/npgs for a list of traits for which data are available). Reports on phenotypic diversity for feed quality traits The USDA-ARS National Small Grains Collection (NSGC) is one of the largest collections of barley germplasm in the world http://www.triticeaecap.org/. Accessed Jan 2014). As part of this USDA-NIFA funded project, 2,417 accessions belonging to the Core were genotyped with a barley iSelect SNP platform, the highest-density genotyping array currently available for barley The Triticeae Coordinated Agricultural Project (TCAP) is focused on genetically and phenotypically characterizing wheat and barley germplasm pools to identify valuable alleles that can be used to develop varieties better adapted to climate change-related stresses  contains 1,860 unique accessions from 94 countries and is comprised of 815 landraces, 781 cultivars/breeding lines, 21 genetic stocks and 243 accessions of undefined improvement status .As shown in http://www.maizegenetics.net. Accessed Jan 2014). The estimated log probability of the data (LnP(D)) for each k between 1 and 10 increased continuously without reaching a plateau . www.ars-grin.gov/npgs. Accessed Jan 2014), we used the geographical centers of the respective country or state/province as the geographic coordinates.Population structure was evaluated using the software STRUCTURE v.2.3.4  plateau .A. To idat k = 2 .B, whichMost accessions within subpopulation 1 exhibit six-row spike morphology and traced to the Mediterranean countries, Australia, and regions of Central and South America . Over haPPD-H2earliness per se locus EPS2VRN-H1VRS1VRS3INT-CRPG4/RPG5 . Further experiments are necessary to prove that a significant association is causal.In the remainder of this report, we will use the following descriptors for significant associations: p-value of the logistic regression of any given marker with another marker located at a specified distance will be, therefore, a direct measurement of LD between the two markers without the confounding effect of population structure. We analyzed the p-value of the logistic regression between any pair of SNPs located 1–2 cM and 4–5 cM apart. In this manner, we systematically scanned along the chromosomes and displayed the extent and distribution of LD in two inter-marker distance intervals. Analyses of LD patterns in the genome can provide insights into recombination hotspots (very low LD) and selective sweeps (high LD) VRS3 on 1H, VRS1 on 2H, INT-C on 4H and NUD on 7H at the 1–2 cM marker distance , high LD was detected at 4–5 cM distance but not at 1–2 cM  decays with genetic distance was investigated for each of the five subpopulations. The patterns of LD decay differ between subpopulations  on chromosome 7HL VRS1 on 2H INT-C on 4H VRS3 on 1H Vrs1per sewww.ars-grin.gov/npgs. Accessed Jan 2014) and are shown in We performed GWAS to test if the natural diversity present in the iCore could be exploited to find genes via association genetics. We chose three phenotypes: ‘hull cover’, ‘spike row number’ and ‘heading date’, which range from monogenic and qualitative (‘hull cover’) to polygenic and quantitative (‘heading date’). ‘Hull cover’ refers to adherence, or lack thereof, of the hull  to the seed and is controlled by a single locus (10(q)  = 140.49) with the most significant marker located at 85.87 cM on chromosome 7H . SNP 11_20606, at 31.14 cM on 4H , based on 4,558 mapped SNPs. In a step-wise process we then identified, one at time, the accessions whose removal from the whole set led to the highest average PIC. If a set of cessions . Choosincessions . As showWe also ranked the accessions by contribution to the average PIC value within the spring two-row (n = 621) and spring six-row (n = 862) accessions . We did Greater genetic diversity than is present in current elite crop varieties will be needed to meet future production goals and the challenges of climate change The USDA-NSGC, the second largest germplasm collection of barley in the world ; (2) accessions may have been chosen based on diversity rather than on representing the principal germplasm group(s) grown at that location; (3) accessions described as landraces may actually be admixed cultivars; and (4) incorrect passport information. Examples of bias in the iCore include, but are not restricted to, the absence of two-row accessions from Australia, where two-row varieties prevail, and Manchurian types from the upper Midwest of the USA, which were the foundational varieties and have been extensively used in breeding programs Although the five subpopulations, in general, correspond to known germplasm groups, care must be taken when considering this collection as a representative sample of barley geographical diversity (present or past) or as a tool to explain origin, domestication or breeding history. There are several reasons why caution is prudent, including: (1) accessions were not always collected Examination of linkage disequilibrium in cultivated barley has been the subject of numerous studies e.g., . To datep-value of a logistic regression between pairs of markers located at a certain distance (between 1–2 cM and 4–5 cM apart), we identified varying degrees of LD across the genome. The fixation of natural mutations by selection can have a large impact on the patterns of LD in nearby regions VRS3 on 1H, VRS1 on 2H, INT-C on 4H and NUD on 7H  family transcription factor nud allele is only present in subpopulation 2 , the mixed linear model properly removed false associations due to structure and we identified significant marker-trait associations at the corresponding chromosome region  the lack of causative SNP(s) because of the ascertainment bias in the iSelect SNP data; and (2) the poor map resolution due to extensive LD found on chromosome 7H in the vicinity of NUD , GWAS narrowed down gene targets for cloned genes to a BAC clone (VRS1) and tight linkage (INT-C). The third gene, VRS3, has not been identified yet but mapping data confirms that it is located near the region identified by our GWAS .The presence or absence of adhering hulls is a simply inherited trait (‘hull cover’) determined by the e region . Althougy of NUD . In the NUD locus are therefore possible candidates for this quantitative variation. ‘Spike row type’ is somewhat more challenging to phenotype than ‘hull adherence’, as there are intermediate types as a result of different VRS1 alleles  loci distributed across the genome ‘Hull adhesion’ and ‘spike row type’ are qualitative traits and the effectiveness of GWAS was validated in that it correctly identified chromosomal regions previously reported to contain loci and/or genes determining each trait. In the case of the ‘hull adhesion’ trait, the two phenotypes are quite discrete and easy to score (e.g. the caryopsis has an adhering hull or it doesn’t). The GRIN data for this trait are binary. There are, however, quantitative differences in the degree of hull adherence within the adhering hull class Heading date, a surrogate measurement of flowering time, is a critical trait in terms of barley adaptation and shows the most complex inheritance of the three traits we studied. At least 6 major genes/QTL are reported to determine heading date in the literature PPD-H1 (SNP within the causative gene), a pseudo-response regulator (PRR7) involved in flowering time under long day conditions EPS2, a homolog of Antirrhinum CENTRORADIALIS which is a main determinant of adaptation in spring barleys VRN-H1 or VRN-H2.The classification of growth habit was made by NSGC curators based on phenotype  . In our a priori selection of a subset of accessions from the whole collection based on a certain criteria (e.g. ‘spike row type’) will lead to different mini-core sets of accessions. It is worth mentioning that, when selecting a ‘mini core set’, researchers should be aware of the ascertainment bias in the iSelect SNP array, which will lead to a higher representation of the breeding materials than landraces. Information regarding subpopulation membership coefficients, geographical origin and phenotypic data provided for each accession  to accelerate the utilization of the genetic diversity contained within the USDA’s remarkable collection of barley accessions.In summary, we have shown that the iCore is a highly diverse collection of barley genetic resources whose effective use will be maximized due to the availability of high density SNP data. The SNP data provide objective criteria for removal of redundant accessions and, as needed, for subsampling ‘mini-core’ sets of accessions for more efficient, cost-effective, or in-depth phenotyping. The high-throughput genotyping data – coupled with a newly developed high-density genetic map – were used to assess patterns of population structure and linkage disequilibrium that we applied to gene discovery using GWAS. For each of three model traits, GWAS identified significant marker-trait associations. The SNPs involved in these associations were in genes known to be responsible for the phenotype, physically linked to determinant genes, and tightly linked to determinant genes/loci. Genotypic and phenotypic data, together with the iSelect consensus map, have been uploaded to T3 . The whole core collection contains 2,574 accessions but only 2,417 were genotyped. Automated SNP calling was performed using the cluster algorithm implemented in GenomeStudio v.2011.1 software . SNP calls were manually inspected to verify their accuracy. Briefly, genotype clusters were manually adjusted for those SNPs with inaccurate cluster definitions, and SNPs producing theta-compressed clusters were excluded. Heterozygous SNP calls were converted to missing values. Standard QC filters were applied to the resultant dataset to remove low-quality SNPs and samples: SNPs with missing calls in >10% accessions as well as accessions with missing calls in >10% of the SNPs were removed from further analysis.The curated SNP dataset consisting of 6,224 SNP markers was used to identify potential duplicates in the NSGC barley core. A pairwise similarity matrix based on simple matching coefficients was built and accessions sharing all alleles were exported. Those genetically identical accessions were also confirmed by DArT markers previously scored in the collection . The 11 linkage maps used by Muñoz-Amatriaín et al. Six markers were present in two different linkage groups and were curated manually. The duplicates for markers 1_0716, 2_1055, 1_0349, and 2_0029 were deleted based on flow-sorting results or synteny in the same manner as Muñoz-Amatriaín et al. k) between 1 and 10, with a burn-in period of 25,000 and 25,000 Monte Carlo Markov Chain (MCMC) iterations. LnP(D) values were plotted and Δk values were calculated according to Evanno et al. k, a new run using a burn-in period of 100,000 and 100,000 MCMC was used to assign accessions to subpopulations based on a membership probability greater than 0.80. Those accessions with a membership probability lower than 0.80 were assigned to an ‘admixed’ group. Principal Component Analysis (PCA) was also conducted in TASSEL v. 3.0 (http://www.maizegenetics.net) on the same dataset.The software package STRUCTURE v.2.3.4 ΦPT for each SNP. We used AMOVA results generated in the GenAlEX 6.5 with 1,000 permutations to estimate ΦPT asAP is the variance among subpopulations and VWP is the variance within subpopulations.Non-admixed accessions from these subpopulations were used to study the population genetic differentiation http://www.maizegenetics.net. Accessed Jan 2014) was used to calculate the linkage disequilibrium (LD) parameter 2r and corresponding p-values (two-sided Fisher’s exact test). For the calculation of LD, markers with minor allele frequency (MAF) below 0.05 and individuals with a percentage of admixture above 80% were excluded. The 2r values were calculated for each chromosome for the different subpopulations and plotted against genetic distance (cM). A hundred thousand pairwise 2r were calculated between randomly selected and physically unlinked markers. The distribution of those 2r values was power transformed to approach normality and the parametric 99th percentile of the distribution was used as a threshold to consider that LD was likely caused by genetic linkage.TASSEL 3.0  was evaluated using adjacent markers, one at a time, as regressors. To assess the variation of p-values as an indirect measure of LD decay at the position M0, we used regressors located at two intervals, between 1 and 2 cM and between 4 and 5 cM from M0.A logistic regression model was used to investigate the relationship between any two markers (binary variables) as a measurement of LD. The analysis was done using SAS v9.3 PROC LOGISTIC . The advantage of this analysis over other traditional measurements of LD - such as Q + K method implemented in TASSEL v. 3.0 as a mixed linear model (MLM) function Q matrix) was accounted for using the result of STRUCTURE for k = 5, and relatedness of accessions was corrected using a kinship matrix (K matrix) generated in TASSEL using SNPs with MAF>0.01. A false discovery rate  were plotted against the consensus genetic position on each chromosome.GWAS was performed on the iCore using the ate FDR;  was usedwww.ars-grin.gov/npgs. Accessed Jan 2014) and are shown in Most of the phenotypic data used for GWAS analyses were obtained from evaluations of the NSGC Barley Core germplasm found at GRIN (Germplasm Resources Information Network system. Available: n) were sorted according to their general contribution to the average polymorphic information content (PIC) of 4,558 mapped SNPs.ip is the frequency of the ith allele k is the No. of alleles.To select a subset of accessions that maximize the diversity of the collection, all the accessions of the collection .In a step wise process, all the accessions were individually removed from the dataset and the average PIC was re-calculated. The accession with the lowest contribution to the PIC  was removed from the next step analysis that starts with Figure S1Exploration of the optimal number of genetic subpopulations (k) in the iCore. (A) Log probability of the data (LnP(D)) for each k between 1 and 10. (B) Δk values calculated as proposed by Evanno et al. k.(TIF)Click here for additional data file.Figure S2Principal Component Analysis (PCA) of the NSGC Barley iCore. The first plot shows the proportion of variance explained by each PC, and the next three plots represent the first four PCs. Accessions are colored by the result of STRUCTURE for k = 5.(TIF)Click here for additional data file.Figure S3Linkage disequilibrium (r2) decay over genetic distance (cM) for the seven barley chromosomes. Significance thresholds are represented as horizontal lines.(TIF)Click here for additional data file.Table S1Potential duplicates in the NSGC Core. Each set of accessions represents potential duplicates based on the SNP information from iSelect genotyping. Accessions marked in yellow were kept, while the rest were discarded.(XLSX)Click here for additional data file.Table S2Information on the 1,860 accessions belonging to the iCore. For each accession, phenotypic and geographic information have been added when available, as well as the subpopulation each accession belongs to (1 to 5) and the proportion of each of the five subpopulations (P1–P5). ‘AD’ indicates admixed individuals.(XLSX)Click here for additional data file.Table S3iSelect consensus genetic map. Two versions of the map are presented: iSelect markers only and all SNPs.(XLSX)Click here for additional data file.Table S4GWAS results of ‘hull cover’, ‘spike row number’, and ‘heading date’.(XLSX)Click here for additional data file.Table S5iCore accessions ranked by their contribution to the average Polymorphism Information Content (PIC) value of the whole set. The reported PIC value of an accession with rank n correspond to the average PIC value of the set with ranks 1 to n-1. The spring 2- and 6-row accessions belonging to the iCore are also sorted by their contribution to the diversity of the corresponding group.(XLSX)Click here for additional data file.Table S6Effect of maximum interval size on total map length and average root mean-squared error (RMSE) between linkage maps and consensus map.(XLSX)Click here for additional data file.


A so-called trace represents a group of solutions which share the same set of reversals that must be applied to sort the original permutation following a partial ordering. By using traces, we therefore can represent the set of optimal solutions in a more compact way. Algorithms for enumerating the complete set of traces of solutions were developed. However, due to their exponential complexity, their practical use is limited to small permutations. A partial enumeration of traces is a sampling of the complete set of traces and can be an alternative for the study of distinct evolutionary scenarios of big permutations. Ideally, the sampling should be done uniformly from the space of all optimal solutions. This is however conjectured to be RA) performs the construction of traces through a random selection of reversals on the list of optimal 1-sequences. The second algorithm (DFALT) consists in a slight modification of an algorithm that performs the complete enumeration of traces. Finally, the third algorithm (SWA) is based on a sliding window strategy to improve the enumeration of traces. All proposed algorithms were able to enumerate traces for permutations with up to 200 elements.We propose and evaluate three algorithms for producing a sampling of the complete set of traces that instead can be shown in practice to preserve some of the characteristics of the space of all solutions. The first algorithm (RA and SWA show a tendency to lose traces with high average reversal length. Such traces are however rare, and qualitatively our results show that, for testable-sized permutations, the algorithms DFALT and SWA produce distributions which approximate the reversal length distributions observed with a complete enumeration of the set of traces.We analysed the distribution of the enumerated traces with respect to their height and average reversal length. Various works indicate that the reversal length can be an important aspect in genome rearrangements. The algorithms  By using this notation, we can represent by a signed permutation the order and the orientation of the genomic markers of one species in relation to those of another.When studying genome rearrangements, we can identify homologous markers with the integers 1,…,interval of a permutation π if there exist i,j∈{1,…,n}, 0 < i ≤ j ≤ n, such that ρ = {|πi|,…,|πj|}, where πx is the element which is in position x of the permutation π. If we write the elements of different intervals in increasing order , we can compare them using lexicographic order.A subset of numbersoverlap if they intersect but none is contained in the other. For example, if π = , then ρ1 = {1,3,4} and ρ2 = {2,3,4} overlap, while ρ3 = {2,3,4,5} and ρ4 = {2,3} do not.Two intervals are said to π and an interval ρ of π, we can apply a reversal on π, that is, an operation which reverses the order and flips the signs of the elements of ρ. If ρ={|πi|,…,|πj|}, then π∘ρ=. Due to this, we can use ρ to denote a reversal.Given a permutation k + 1 permutations can be represented by a sequence π0,π1,…,πk of permutations. In such a sequence, every pair of consecutive permutations πi and πi + 1)((0 ≤ i < k) are just one reversal apart ( or vice-versa).A sequence of ρ1…ρd sorts a permutation π0 if ρi is an interval ofi, andπd ). A shortest sequence of reversals sorting a permutation is called an optimal sorting sequence. The length of such a sequence of reversals is called reversal distance, denoted by d (or simply d(π0), whenπ0= andd=d(π0)=4.A sequence of reversals Sorting Permutations by Signed Reversals (SPSR) and has been the topic of a vast literature. The first polynomial algorithm with complexity O(n4) was proposed in 1999 by Hannenhali and Pevzner[The problem of finding an optimal sorting sequence under this model (henceforward denoted by HP) due to Hannenhalli and Pevzner is calle Pevzner. In 2001 Pevzner. In 2004n3/2logn).et al.[k is the number of successive corrections which must be applied when the algorithm chooses an unsafe reversal. Swenson et al. showed a permutation family where k is Λ(n) (worst-case for k) and, in this case, the algorithm is quadratic. However, tests performed by the authors showed that k generally is a constant smaller than 1 and independent of the permutation size. Because of this, the algorithm has, with high probability, execution timeMore recently, Swenson et al. proposedO(nlogn).Double Cut and Join operation (DCJ). It allows the study of evolutionary scenarios between genomes which are composed by one or more chromosomes, either linear or circular. This universal operation accounts for reversals, translocations, fusions and fissions. Transpositions and block interchanges are modelled by two DCJ operations. Bergeron, Mixtacki and Stoye gave a linear time algorithm to compute the DCJ distance between two genomes[DCJ and developed an algorithm to count the number of optimal sorting sequences[DCJ sorting sequence can be obtained from another one by applying replacements of consecutive operations. However, the problem of finding the shortest number of replacements is still open and an algorithm to enumerate all DCJ rearrangement scenarios is currently not available. Furthermore, the DCJ model appears less relevant then the classical HP model as it allows for mutations that are rarely or never observed in biological data (e.g. circular chromosomes for eukaryotes)[A more general evolutionary model than the HP model was proposed by Yancopoulos, Attie and Friedberg called t genomes. Braga aequences. Additioaryotes).SPSR algorithms for the HP model however output just one optimal sequence of reversals, while the space of optimal solutions can be huge. Hence, the solution produced by such algorithms may differ from the scenario which really occurred during the evolution of the genome, even when such scenario indeed verifies some optimality criterion. For instance, the permutation  has 6345019 optimal solutions.The traditional π0 and a target permutation πd, an optimal 1-sequence is a reversal that, when applied to π0, produces a permutation π1such that d = d−1. In the same way, an optimal i-sequence is a sequence of i reversals that, when applied to π0, produces a permutation πisuch that d = d−i.Given a permutation optimal 1-sequences of a given permutation in time O(n3)[d(π0πd)-sequences that sort the permutation π0into πd. Recently, Swenson, Badr and Sankoff presented a quadratic algorithm to enumerate the optimal 1-sequences of a permutation[In 2003, Siepel proposed an algorithm which calculates the set of all me O(n3). It is tmutation.i.e. they do not require the sorting sequence to have minimum size). This was extended in 2004 to include translocations[York, Durret, and Nielsen proposed in 2002 a Bayesian approach for the problem of inferring the history of inversions which separate the homologous chromosomes from two different species. The metocations. In 2003ocations.et al. were designed to infer the sequence of rearrangement events that explain the difference between two species. Larget et al. developed a method to analyse the complete mitochondrial genome rearrangements of 87 metazoa taxa[BADGER.The methods of Mikls and York zoa taxa,17. TheiMC4Inversion. It uniformly samples the set of all optimal paths and can estimate the total number of optimal sorting paths.An MCMC approach was also proposed by Mikls and Darling in 2009 for sampling parsimonious reversal histories. The metet al. introduced the concept of traces for the SPSR problem. This concept allows the organisation of a set of optimal solutions into classes[et al. define a relation over them: if ρ and θ are reversals  and they show no overlap, then the words ρθ and θρ are said to be equivalent. We say that ρ and θcommute. Based on this relation, any word that contains the subword ρθ is equivalent to the same word obtained by replacing ρθ by θρ. For example, the sequences of reversals (words) {1}{1,2,3}{2,3,4} and {1,2,3}{1}{2,3,4} are equivalent because the reversals {1} and {1,2,3} commute. Inversely, none of these sequences of reversals is equivalent to {1}{2,3,4}{1,2,3} because the reversals {1,2,3} and {2,3,4} overlap.Bergeron  classes. If sequtrace. Bergeron et al. proposed that for a given signed permutation π, the set of all optimal solutions is a union of traces. Thus, traces can be used to produce a more relevant result to the SPSR problem because they provide a more compact representation of an enormous set of solutions.A class of optimal reversal sequences over this relation is called a s of a trace T is in its normal form if it can be decomposed into subwords s = u1|…|um such that: An element • every pair of elements of a subword uicommute;• for every element ρof a subword ui(i > 1), there is at least one element θ of the subword ui−1such that ρand θ do not commute;• every subword uiis a nonempty increasing word under the lexicographic order.A theorem by Cartier and Foata states that, for any trace, there is a unique element that is in normal form. This alheight. The size of a trace T is the number of solutions which it represents. The trace {1,2,4}{3}|{1,3,4}|{2,3,4} has height 3 and size 4 because it represents just the solutions: {3}{1,2,4}{1,3,4}{2,3,4}, {1,2,4}{3}{1,3,4}{2,3,4}, {1,2,4}{1,3,4}{3}{2,3,4}, and {1,2,4}{1,3,4}{2,3,4}{3}.The number of subwords in a trace denotes its T that contains an optimal sequence ρ1…ρd can be represented by a partial ordering of the setρi < Tρj if and only if ρi is always before ρj in the elements of T. For example, in the trace T = {1,2,4}{3}|{1,3}|{3,4}, the elements ofT {1,3}, {1,3}<T {3,4}, and {1,2,4}<T {3,4}. Notice that the reversal {3} is not comparable to the others by the relation <T because, given that it does not overlap the other reversals, it can be placed in the sequence before or after any of them.A trace T form a partially ordered set (poset). The width of a trace (or poset) is a maximum cardinality set of elements ofT. It is at least  the maximum size of a subsequence ui in the normal form of a trace.The setπ0 =  into ι4). A node of the tree represents a set of reversals that is sorted in lexicographic order. The root node contains the optimal 1-sequences of the original permutation. To each reversal ρ of a non-leaf node is attached a subtree which groups the reversals that are lexicographically bigger than it or that should be applied after it. For example, in FigureA contains the optimal 1-sequences of π0 and the reversal {1} of node A has a subtree, rooted at node B, attached to it. Node B contains the optimal 1-sequences of the permutation π1 obtained after applying the reversal {1} over π0.A set of solution traces can be represented using a sorted tree similar to the one shown in Figure0 = −3,2,,−4 into i of the tree gives us an i-trace. An i-trace represents a set of solutions which sort πk into πk + i), then we have a trace that sorts the permutation π0 into πd.Every path from the root to a node at level et al. combined Siepel’s algorithm with the concept of traces[Braga f traces and deveConsidering some biological criteria, constraints can be applied in the selection of the reversals during the process of enumeration. Thus, the size of the output can be reduced,22. For et al. explores the tree of solution traces in breadth-first manner and adopts a complex data structure to keep the intermediary information into the main memory and disk[Breadth-First Algorithm or BFA for short.The algorithm proposed by Braga and disk. Due to Depth-First Algorithm (or DFA for short) makes use of a stack structure to keep the intermediary data on the main memory only. With this solution, we greatly reduced the amount of data which must be kept in the main memory and we eliminated the disk accesses. However, this algorithm cannot be used with most of the biological constraints developed by Braga et al., and cannot compute the total number of solutions that is represented by the set of traces.In a previous work, instead of exploring the universe of solutions in breadth-first manner, we adopted a depth-first strategy to explore the branches of the tree of solution traces. This DeBFA algorithm iskmax is the maximum width of a trace and N is the number of traces of solutions[DFA algorithm is O(Nn42n)[The time complexity of the olutions. The comO(Nn42n).i-traces according to the permutation that they produce when their sequences of reversals are applied to the original permutation. As many traces can produce the same intermediary permutation, by grouping them, the authors avoid unnecessary computations. Instead of generating the set of optimal 1-sequences for every i-trace, they compute this set just for the intermediary permutation which groups a set of i-traces. Despite the gain of 70% over the execution time of the BFA algorithm and 50% over the DFA algorithm, the methods proposed by Badr, Swenson and Sankoff use a considerable amount of the main memory to keep the groups of i-traces and permutations.Recently, Badr, Swenson and Sankoff adapted the two algorithms of trace enumeration. The strn≥15), the time necessary to produce the complete set of traces makes impracticable any analysis.Although sets of traces are smaller than their equivalent sets of solutions, the number of traces also increases exponentially with the size of the permutations and their reversal distance. Thus, for big permutations  is very simple and constructs the traces through a random selection of reversals in the list of optimal 1-sequences. The second algorithm (DFALT) represents a slight modification of the DFA algorithm. Finally, the last algorithm (SWA) is more elaborate and makes uses of a sliding window strategy to improve the enumeration of traces.The three algorithms were designed to enumerate traces while a given execution time limit is not reached. The first ), the algorithm applies each (i−1)-trace t on π0 to produce a new permutation πi−1)( (i.e. πi−1)( = π0∘t). The list of optimal 1-sequences of πi−1)-trace t to generate a new set of i-traces.Then at each iteration i=d, all traces which sort π0into πdhave been enumerated.When the algorithm finishes to process level A depth-first strategy was adopted by Baudet and Dias to explore the tree of solution traces.i-trace.The algorithm makes use of a stack structure to handle the data produced during the process. Each level of the stack has a list of reversals sorted in lexicographic order. Additionally, the sequence of reversals constructed with the first reversal of each level, from the bottom to the top of the stack, represents the current π0 is pushed into the first level of the stack. While the stack is not empty, the algorithm gets the current i-trace t and applies it to the permutation π0 to produce the permutation πi. Each reversal ρ of the list of optimal 1-sequences of πi is added to the list that will be pushed into the top of the stack only when it is the last reversal of the (i + 1)-trace t″=t + ρ. If the reversal ρ does not appear in the last position of the (i + 1)-trace t″, it means that it belongs to another branch of the tree of solution traces and, therefore, it can be ignored.First, the list of optimal 1-sequences of the permutation i = d, the algorithm outputs the current trace t and removes from the list the reversal that is on the top of the stack. Every time the top of the stack contains an empty list, the algorithm pops it and removes the first reversal of the list that is in the new top. The algorithm finishes when the stack becomes empty.When the stack reaches the level π0into πd, different i-traces can sort π0 into the same intermediary permutation πi. Based on this observation, Badr, Swenson and Sankoff adapted the BFA and DFA algorithms to speed-up the trace enumeration[During the enumeration of traces which sort meration.i-traces according to the permutation that they produce when their sequences of reversals are applied to the original permutation. Instead of computing sets of optimal 1-sequences for every i-trace, this procedure is performed only for each intermediary permutation which appears on the level i.The strategy consists in grouping BFA and DFA.Tests performed by the authors show that, on average, this change in the algorithms results in a gain of 70% and 50%, respectively, over the total execution time of the algorithms Due to the exponential nature of the set of solutions, the algorithms that sort all solution traces are not suited for processing big permutations.With the objective of calculating alternative evolutionary scenarios for big permutations, we developed three different algorithms that perform a partial enumeration of the complete set of solution traces of a given permutation.The proposed algorithms output traces while they do not reach the stopping criterion which, in our case, is a given execution time limit.A very simple solution for the partial enumeration of traces is to construct random traces.π0 be the original permutation and πd the target permutation. This method consists in generating a trace through the random selection of a reversal among those in the set of optimal 1-sequences of each permutation πi which is between π0and πd). This process is repeated while the execution time limit is not reached.Let DFA explores the tree of solution traces branch by branch. Moreover, every time it reaches a leaf node, it outputs a new trace. Consequently, another simple alternative to producing a set of traces is to use the algorithm DFA and introduce a verification over the elapsed time to interrupt its execution when the time limit is reached.The algorithm BFA. As it outputs the enumerated traces only when it reaches the last level, the necessary time limit to output at least one trace would be very close to the time required to enumerate all traces.Observe that the same procedure cannot be adapted to the algorithm πk be an intermediary permutation that is obtained after applying the first k reversals of an optimal sequence of reversals which transforms π0into πd). In this context, we can define the k-trace X and the l-trace Y , where l = d−k. X and Y  are, respectively, the traces which represent all solutions that transform π0 into πk and, πk into πd.Let Y  and add each one of them, sequentially, to X, we produce a trace Z that transforms π0 into πd. For example, Figureπ0 =  into the permutation π8 = . The 4-trace A = {1,−,6,8}{1,−,3,5,6}{8}|{1,−,7} represents a solution trace which transforms π0 into the intermediary permutation π4. In the same way, the 4-trace C = {2,5,6}{3,4}{6}|{3,−,6} represents a solution trace which transforms π4 into π8. By adding each reversal of C into the trace A, we build the 8-trace AC = {1,−,6,8}{1,−,3,5,6}{2,5,6}{6}{8}|{1,−,7}{3,4}|{3,−,6} which sorts π0into π8.If we get the reversals of w that produces sets of k-traces (1 ≤ k ≤ w) which transform: π0 into π1, π0 into π2, …, π0 into πw−1) can be combined in the following way: After that, these sets of • 1-traces that transform π0 into π1 with w-traces that transform π1 into π(w + 1) to generate (w + 1)-traces which transform π0 into π(w + 1);• 2-traces that transform π0 into π2 with w-traces that transform π2 into π(w + 2) to generate (w + 2)-traces which transform π0 into π(w + 2);• …• (d−1)-traces that transform π0 into π(d−1)with 1-traces that transform π(d−1)into πdto generate d-traces which transform π0into πd.RA to return the list of all intermediary permutations .The first step of this algorithm consists in generating a random set of intermediary permutations. To do this, we can adapt the algorithm k-traces (1 ≤ k ≤ w) that transform πi into πi + k).In our example of FigureBFA was implemented in Java by Braga[DFA in order to adopt the same Java Objects that were used by Braga. The algorithms RADFALT, and SWA were also implemented under the same Java package structure.The algorithm by Braga. Startin-Xmx1024m).The tests were performed on an Intel Pentium 4 HT 3.0 GHz with 2.0 GB of RAM running Ubuntu. To avoid the influence of swap operations on the performance of the structures, we limited the maximum memory that the Java Virtual Machine could allocate to 1.0 GB .During the tests we collected the maximum amount of main memory used by the algorithms. The memory was measured through a separated thread that at regular intervals collected the memory used by the Java Virtual Machine <d or πi−1), we evaluated the average time which is necessary to enumerate all traces of a given permutation. The objective here is to show the exponential behaviour of the sizes of the sets of traces, depending on the number of elements of the permutations and their reversal distances, and to collect information to determine the time limits that will be used to evaluate the proposed algorithms. We shall see later how to choose such a time limit in general, or some other stop criterion.Before performing comparative tests with the three proposed algorithms  = 13) required three days to be processed. For each permutation, we enumerated all traces with the algorithm DFA, and we collected the execution time. For each set of permutations, we calculated the average number of traces and the average execution time. FiguresWe created sets of 500 random linear and circular permutations with, respectively, 10 and 15 elements with reversal distances between 4 and 13. These values were chosen because they allow the total enumeration of traces, for the set of 500 permutations, in a reasonable time. For example, just the set /n. For example, when we fix the value 9 for the reversal distance, permutations with 10 elements have on average more traces than permutations with 15 elements. The same observations made for FigureFigureThe average amount of traces observed for circular permutations is bigger than the one observed for linear permutations with the same reversal distance. For every reversal in a linear permutation, there exists two equivalent reversals in the corresponding circular permutation. Thus, circular permutations have a higher number of optimal solutions than linear permutations. This characteristic is indicated by the curves of Figuren = 15, d(π) = 12) which leads to an average execution time of, approximately, 30 seconds.To evaluate the proposed algorithms, we decided to adopt a set of permutations which lead to an average execution time that is neither too short, nor too long. As the behaviour of linear and circular permutations are similar, we opted for performing tests only with linear permutations. Based on these criteria, we chose the set . We also counted the number of traces that have average reversal length R(2≤R≤11).First, for each permutation P was processed by each pair . For each of these executions, we calculated the percentage of all traces of height H  of the permutation P that we sampled with the algorithm A within the time limit T.Considering that the average time to process the permutations of the selected set is 30 seconds, we used the algorithms with the following time limits: 6, 12, 18, 24, 30, and 36 seconds. Each permutation A, T), we calculated for the set of 500 random permutations the average percentage of all traces of height H  that were enumerated by A within the time limit T. The plots in FiguresH and R respectively.Finally, for each pair  is just 30 seconds. It is a set of permutations whose traces can be easily enumerated. Nonetheless, these algorithms were developed with the objective of enumerating traces of big permutations which demand a huge processing time.The average time to process this set of permutations (d = ⌈(n + 1)/2⌉. Each permutation was processed by each proposed algorithm with a time limit of 60 seconds. For each execution, we collected the number of enumerated traces and the maximum amount of memory used by the algorithm. Figuresn respectively, the average number of enumerated traces and the average memory usage observed during the executions of each set of 100 permutations.To check whether they were capable of doing this, we created sets of 100 random permutations with a number of elements varying between 40 and 200 and a reversal distance DFALT decreases as the size of the permutations increases. This phenomenon is associated with the time that this algorithm spends processing dead branches in the tree of traces.We can see in FigureRA has a curve very similar to the one shown by DFALT but its shape has a different explanation. When we increase the number of elements of the permutation and, consequently, the initial reversal distance, we have that the time that is spent on the analyses of the breakpoint graph to find an optimal 1-sequence also grows. Because of this, the number of traces which are enumerated by the algorithm RA decreases when we increase the initial reversal distance of the permutations.RA lose in performance does not affect the SWA algorithm. Even with an increment of the initial reversal distance, SWA is all the time concerned with the enumeration of k-traces (1 ≤ k ≤ w). As w is usually small, SWA does not lose in performance when producing optimal 1-sequences.The reason that makes the algorithm k-traces that transform πi into π(i + k). Because of this, we profit from all the structures that are created for the generation of the optimal 1-sequences. In the case of the algorithm RA, even if we avoid to generate all structures, the created ones are partially explored because just one reversal is considered for each intermediary permutation.Another advantage of the sliding window strategy is that it produces all SWA is able to enumerate more traces than the other two algorithms when the same time limit is imposed. For permutations with up to 120 elements, the algorithm SWA6 enumerates the highest number of traces. For bigger permutations, the algorithm SWA5 outperforms the algorithm SWA6.FigureSWA5 and SWA6 present curves that have a parabolic shape. The number of enumerated traces grows up to a given point, and then starts to decrease. The explanation for this behaviour lies in the process of combining the i-traces and the k-traces. When we combine xi-traces with yk-traces, we can create up to x×y(i + k)-traces (some of the generated traces can appear more than once). Thus, if the reversal distance of the original permutation increases, the number of combinations (i-traces + k-traces) and the time that is spent on them also increases.We can see in FigureSWA6 initially enumerates many more traces but the reduction in the performance also starts earlier than for the other two tested values of window.Generally, a set of 6-traces is bigger than a set of 5-traces and much bigger than a set of 4-traces. As a consequence, we can see that the algorithm RA and DFALT have a small variation in the average memory and that the algorithm SWA consumes more memory. While SWA4 has a more stable memory usage, SWA5 and SWA6 have an ascending curve of memory usage.FigureDFALT outputs every trace just once. Because of this, when using DFALT, we can print the traces avoiding to keep them in memory with the purpose of controlling duplicated traces.While the random algorithms can eventually produce the same trace more than once, RA is related to the interval where it outputs more traces. When the number of enumerated traces decreases, the amount of space that we need to keep the traces in memory also diminishes. As a consequence, memory consumption reaches a level that is low enough for the maintenance of the objects which are being used to produce the enumeration.The higher memory usage of SWA, we have to keep in memory the sets of traces which were enumerated and the set of traces which are going to be combined. To reduce memory consumption, we could print all enumerated traces but, as a result of this, we must add a post-processing step to eliminate the duplicate traces.In the case of the algorithm ♯P-complete[When we perform a sampling of a big set of elements, usually we must verify whether the result is unbiased. This implies checking if the output of the algorithms covers the space of solutions uniformly. This task has been conjectured to be complete. We thern = 15 and d = 12 and, for each permutation P, we calculated the ratios  and , using the complete set of traces  and the outputs of the executions of each algorithm. For each permutation, we thus have the distribution of its traces according to the height, and to the average reversal length of the traces.We considered the set of 500 random permutations with A, time limit T) and for the set Total, we calculated the average ratio for each value of H and R over all 500 permutations. FiguresFor each pair , we generated for the set Total and for each pair  distributions of 5000 values  respecting the average ratios observed on the 500 permutations. These generated distributions were compared with the statistical test.Finally, to compare the distributions of height and reversal values obtained for the set DFALT, 36s),  and  were considered to be similar  to the distribution of the set Total with p-values, respectively, equal to 0.94, 0.46 and 0.04. In the case of the parameter average reversal length, the distributions produced for the pairs , ,  and  were considered to be similar  to the distribution of the set Total with p-values, respectively, equal to 0.71, 0.25, 0.03 and 0.02.For the parameter height, the distributions produced for the pairs  depending on the time limit imposed. The algorithm RA enumerates less traces and, consequently, is more distant from the reference curve. On the other hand, the algorithm DFALT enumerates more traces and has the best approximation to this curve. Considering the executions of the algorithm SWA, we have that SWA6 is the one that is closer to the reference curve.The curves of the proposed algorithms exhibit different levels of approximation to the reference curve  approximates the reference curve, with a p-value of 0.02.In the case of the non-deterministic algorithms, we cannot guarantee that the sampling will have the same property for small or big permutations. Nonetheless, FiguresRA, DFALT, and SWA. Designed for processing big permutations, all proposed algorithms are able to do a partial enumeration of traces for permutations which cannot be processed by the actual algorithms for total trace enumeration, that is, BFA and DFA with or without permutation grouping.In this work, we proposed three different algorithms for the partial enumeration of traces: DFALT and SWA are based on the algorithm DFA. Thus, they inherited the inability of working with most of the biological constraints implemented by Braga et al.[RA can be easily adapted to consider these constraints.The algorithms ga et al.,22. HoweSWA is capable of producing a number of traces higher than the ones produced by the other two algorithms when the same time limit is imposed during the processing of big permutations.Among the three proposed solutions, the algorithm During our tests, we worked with time limits between 6 and 60 seconds. However, it is not an easy task to determine the time limit which should be used to produce a good sampling of the total space of traces which sort a given permutation.One alternative could be to adopt other types of stopping criteria. For example, the algorithm could stop after achieving a fixed number of enumerated traces, or a fixed number of repeated traces . Nevertheless, these kind of criteria would be subjected to the same problems as the time limit criterion with respect to the guarantee of obtaining a uniform sampling.A more advanced solution could involve a detailed analysis of the space of solutions of traces to determine a way of calculating the expected total number of traces. As a by-product, we could determine a percentage of the expected number of traces and use it as a stopping criterion.d/n. Nevertheless, to predict the number of traces which sort a permutation is an open question which requires more investigation. A deeper study about the characteristics of the permutations must be conducted with the aim of obtaining, if possible, a formula to calculate the expected number of traces of a permutation. In this direction, the results obtained by Braga and Stoye when analysing the solution space of sorting by DCJ operations[In our tests, we could observe that the number of traces grows exponentially according to the ratio erations may provMC4Inversion, written by Mikls and Darling[d(π0πd)!. In this estimation, we assume that every trace contains only non-overlapping reversals. Obviously, this is not true and, in fact, the real number of traces can be much higher than this estimated lower bound.The difficulty of analysing the quality of the sampling for big permutations is that, for now, we are capable neither to calculate the expected total number of traces, nor to predict the general distribution of the complete set of traces just by looking at the permutation. The software  Darling, could bDFALT and SWA have distributions that get closer to the distribution observed for the complete set of traces when we increase the time limit. This may give an indication that these algorithms can produce unbiased sets of traces, at least in relation to the distribution of height and average reversal length.We conducted tests with small permutations and we could see that the sets of traces partially enumerated by the algorithms The height of a trace does not have a direct biological meaning but it provides some evidence of the complexity of the solutions that it represents. Traces with high height group solutions that have a high number of reversal overlaps. For example, some groups of bacteria evolve mainly through symmetrical or almost-symmetrical reversals relatively to the replication terminus. In this case, we could expect that the occurrence of small reversals contained inside big ones exhibits a ratio bigger than the one observed when the position of the reversals are not restricted. As a consequence, we would then also expect the solution traces to have low height values.RA and SWA show a tendency for losing traces that have high average reversal length. However, if we know that the target genome is subjected to reversals of small or intermediate sizes[The average reversal length can be an important aspect in genome rearrangements. The algorithms te sizes-29, the Independently of a biological meaning, the parameters height and average reversal length represent measures that are easy to compute and that can be used in the evaluation of the quality of a sampling.The authors declare that they have no competing interests.Implementation and test analyses were performed by CB. All authors participated in the discussions. The manuscript was written by CB with major contributions by MFS and ZD. All authors read and approved the final manuscript.