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CMLL is a step that is pretty similar to COLL, but with one key difference: it does not care if the M slice is solved. This means that algorithms can be shorter. However, it is possible to use COLL algorithms for CMLL, but not the other way around. There are 42 algorithms, and it is recommended that you learn all of them, but you can also just use sune and anti sune to orient the corners, then use Jb and Y perms to position them.
All O group cases have all the corners pre-oriented, all H and Pi group cases have no corners oriented; in H they are arranged in two sets of headlights, one facing away and the other facing towards you, and in Pi they are arranged with a set of headlights facing left or right and the other two corners facing away from each other. All U, T, and L cases have 2 corners oriented, and all S and As cases have only one corner oriented.
The original Roux proposal was to orient all the centers and edges, then to permute the UR and UL edges then finally permute the M slice edges. The way that is probably easiest for beginners is to solve the F and B centers, then insert the DB and DF edges, then orient all the U layer edges, then permute them. It is quite inefficient though.
If there are two opposite edges oriented, then hold them in UF and UB and do M' U' M U M' U2 M U' M' U2 M U M' U' M to turn it into the case where there are two adjacent edges oriented.
We present RaGOO, a reference-guided contig ordering and orienting tool that leverages the speed and sensitivity of Minimap2 to accurately achieve chromosome-scale assemblies in minutes. After the pseudomolecules are constructed, RaGOO identifies structural variants, including those spanning sequencing gaps. We show that RaGOO accurately orders and orients 3 de novo tomato genome assemblies, including the widely used M82 reference cultivar. We then demonstrate the scalability and utility of RaGOO with a pan-genome analysis of 103 Arabidopsis thaliana accessions by examining the structural variants detected in the newly assembled pseudomolecules. RaGOO is available open source at
Another reference-free method for pseudomolecule construction involves the use of long-range genomic information to scaffold assembled contigs. This includes a large class of technologies such as mate-pair sequencing, Bacterial Artificial Chromosomes (BACs), Linked Reads and chromatin conformation capture such as Hi-C [6,7,8]. In particular, Hi-C has recently been shown to be a practical and effective resource for chromosome-scale scaffolding [9,10,11]. Paired-end Hi-C sequencing reads are aligned to the assembly, and mates which align to different contigs (Hi-C links) are recorded. According to the relative density of such Hi-C links between pairs of contigs, contigs can be ordered and oriented into larger scaffolds, potentially forming chromosome-length pseudomolecules. Also, because misassemblies may be observed by visualizing Hi-C alignments, Hi-C can be used for validation and manual correction of misassemblies [12]. Though Hi-C has been widely adopted, there remain challenges that can impede the ability to form accurate chromosome-scale pseudomolecules with Hi-C alone. Principally, Hi-C data are noisy, and Hi-C-based scaffolders are prone to producing structurally inaccurate scaffolds [13]. Also, because this process relies on the alignment of short Hi-C sequencing reads to the draft assembly, small and repetitive contigs with little or conflicting Hi-C link information often fail to be accurately scaffolded. Finally, the analysis requires deep sequencing coverage and therefore can be expensive and compute-intensive.
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Here, we introduce RaGOO, an open-source method which utilizes Minimap2 [24] alignments to a closely related reference genome to quickly cluster, order, and orient genome assembly contigs into pseudomolecules. RaGOO also provides the option to correct apparent chimeric contigs prior to pseudomolecule construction. Finally, structural variants (SVs), including those spanning gap sequence, are identified using an optimized and integrated version of Assemblytics [25], thus enabling rapid pan-genome SV analysis of many genomes at once. This is especially important for detecting large insertions and other complex structural variations that are difficult to detect using read mapping approaches.
We first demonstrate the speed and accuracy of RaGOO scaffolding with simulated data of increasing complexity and show that it outperforms 2 popular alternative methods. We next show the utility of RaGOO by creating high-quality chromosome-scale reference genomes for 3 distinct wild and domesticated genotypes of the model crop tomato using a combination of short and long-read sequencing. Finally, we demonstrate the scalability of RaGOO by ordering and orienting 103 draft Arabidopsis thaliana genomes and comparing structural variants across the pan-genome. This uncovers a large number of defense response genes that are highly variable.
The RaGOO pipeline. a Contigs are aligned to the reference genome with Minimap2 and are ordered and oriented according to those alignments. b Normal alignments between a contig and a reference chromosome (top) and example alignments between a reference chromosome and an intrachromosomal chimera (bottom left) and an interchromosomal chimera (bottom right). Red arrows represent potential contig breakpoints
We have introduced RaGOO in both a general and focused context for highly accurate genome scaffolding. As a general method, RaGOO may be valuable for chromosome-scale scaffolding in experimental designs where ordering and/or orienting of contigs leveraging an existing reference is available. Ordering and orienting with RaGOO may also facilitate analysis not possible with unlocalized contigs. This is exemplified by the additional sequence found through gap filling of the M82, BGV, and FLA assemblies or by the identification of structural variants spanning gaps between contigs in the S. lycopersicum and Arabidopsis thaliana pan-genomes. Additionally, our pan-genome analysis demonstrates that the speed of RaGOO offers new possibilities as to the scope and size of experiments that require reference-guided scaffolding. Furthermore, the integrated structural variant identification pipeline allows for a rapid survey of gene-related and other variants in the population. This shows that for both tomato and Arabidopsis pan-genomes, the majority of protein-coding genes are associated with the structural variation, highlighting the importance of population-scale assembly and structural variant discovery.
Each contig is assigned a confidence score, between 0 and 1, for each of the three stages outlined above. The clustering confidence score is the number of base pairs a contig covered in its assigned reference chromosome divided by the total number of covered base pairs in the entire reference genome. To create a metric associated with contig ordering confidence, we defined a location confidence. First, the smallest and largest alignment positions, with respect to the reference, between a contig and its assigned reference chromosome are found. The location confidence is then calculated as the number of covered base pairs in this range divided by the total number of base pairs in the range. Finally, to calculate the orientation confidence, each base pair in each alignment between a contig and its assigned reference chromosome casts a vote for the orientation of its alignment. The orientation confidence is the number of votes for the assigned orientation of the contig divided by the total number of votes.
Prior to clustering, ordering, and orienting, RaGOO provides the option to break contigs which may be chimeric as indicated by discordant alignments to the reference. RaGOO can identify and correct both interchromosomal and intrachromosomal chimeric contigs. Interchromosomal chimeric contigs are contigs which have significant alignments to two distinct reference chromosomes. To identify and break such contigs, all the alignments for a contig are considered. Alignments less than 10 kbp are removed, and the remaining alignments are unique anchor filtered [25]. If there are multiple instances where at least 5% of the total alignment lengths cover at least 100 kbp of a distinct reference chromosome, a contig is deemed chimeric. To break the contig, alignments are sorted with respect to the contig start, then end positions, and the contig is broken where the sorted alignments transition between reference chromosomes.
We recorded various measurements to evaluate the success of these tools in ordering and orienting simulated assemblies. Firstly, we observed the runtime, percentage of localized contigs, and percentage of localized sequence. To assess the clustering and orienting accuracy, we measure the percentage of localized contigs that had been assigned the correct cluster and orientation, respectively. Finally, we used two measurements to assess the ordering accuracy of each pseudomolecule. The first was the edit distance between the true and predicted order of contigs. This edit distance was normalized by dividing by the total number of contigs in the true ordering. The second ordering accuracy measurement was the percentage of correct adjacent contig pairs.