Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.Comment: 19 pages, 4 figure

Comprehensively Surveying Structure and Function of RING Domains from Drosophila melanogaster

Using a complete set of RING domains from Drosophila melanogaster, all the solved RING domains and cocrystal structures of RING-containing ubiquitin-ligases (RING-E3) and ubiquitin-conjugating enzyme (E2) pairs, we analyzed RING domains structures from their primary to quarternary structures. The results showed that: i) putative orthologs of RING domains between Drosophila melanogaster and the human largely occur (118/139, 84.9%); ii) of the 118 orthologous pairs from Drosophila melanogaster and the human, 117 pairs (117/118, 99.2%) were found to retain entirely uniform domain architectures, only Iap2/Diap2 experienced evolutionary expansion of domain architecture; iii) 4 evolutionary structurally conserved regions (SCRs) are responsible for homologous folding of RING domains at the superfamily level; iv) besides the conserved Cys/His chelating zinc ions, 6 equivalent residues (4 hydrophobic and 2 polar residues) in the SCRs possess good-consensus and conservation- these 4 SCRs function in the structural positioning of 6 equivalent residues as determinants for RING-E3 catalysis; v) members of these RING proteins located nucleus, multiple subcellular compartments, membrane protein and mitochondrion are respectively 42 (42/139, 30.2%), 71 (71/139, 51.1%), 22 (22/139, 15.8%) and 4 (4/139, 2.9%); vi) CG15104 (Topors) and CG1134 (Mul1) in C3HC4, and CG3929 (Deltex) in C3H2C3 seem to display broader E2s binding profiles than other RING-E3s; vii) analyzing intermolecular interfaces of E2/RING-E3 complexes indicate that residues directly interacting with E2s are all from the SCRs in RING domains. Of the 6 residues, 2 hydrophobic ones contribute to constructing the conserved hydrophobic core, while the 2 hydrophobic and 2 polar residues directly participate in E2/RING-E3 interactions. Based on sequence and structural data, SCRs, conserved equivalent residues and features of intermolecular interfaces were extracted, highlighting the presence of a nucleus for RING domain fold and formation of catalytic core in which related residues and regions exhibit preferential evolutionary conservation

A Single Molecule Scaffold for the Maize Genome

About 85% of the maize genome consists of highly repetitive sequences that are interspersed by low-copy, gene-coding sequences. The maize community has dealt with this genomic complexity by the construction of an integrated genetic and physical map (iMap), but this resource alone was not sufficient for ensuring the quality of the current sequence build. For this purpose, we constructed a genome-wide, high-resolution optical map of the maize inbred line B73 genome containing >91,000 restriction sites (averaging 1 site/∼23 kb) accrued from mapping genomic DNA molecules. Our optical map comprises 66 contigs, averaging 31.88 Mb in size and spanning 91.5% (2,103.93 Mb/∼2,300 Mb) of the maize genome. A new algorithm was created that considered both optical map and unfinished BAC sequence data for placing 60/66 (2,032.42 Mb) optical map contigs onto the maize iMap. The alignment of optical maps against numerous data sources yielded comprehensive results that proved revealing and productive. For example, gaps were uncovered and characterized within the iMap, the FPC (fingerprinted contigs) map, and the chromosome-wide pseudomolecules. Such alignments also suggested amended placements of FPC contigs on the maize genetic map and proactively guided the assembly of chromosome-wide pseudomolecules, especially within complex genomic regions. Lastly, we think that the full integration of B73 optical maps with the maize iMap would greatly facilitate maize sequence finishing efforts that would make it a valuable reference for comparative studies among cereals, or other maize inbred lines and cultivars

Detailed Analysis of a Contiguous 22-Mb Region of the Maize Genome

Most of our understanding of plant genome structure and evolution has come from the careful annotation of small (e.g., 100 kb) sequenced genomic regions or from automated annotation of complete genome sequences. Here, we sequenced and carefully annotated a contiguous 22 Mb region of maize chromosome 4 using an improved pseudomolecule for annotation. The sequence segment was comprehensively ordered, oriented, and confirmed using the maize optical map. Nearly 84% of the sequence is composed of transposable elements (TEs) that are mostly nested within each other, of which most families are low-copy. We identified 544 gene models using multiple levels of evidence, as well as five miRNA genes. Gene fragments, many captured by TEs, are prevalent within this region. Elimination of gene redundancy from a tetraploid maize ancestor that originated a few million years ago is responsible in this region for most disruptions of synteny with sorghum and rice. Consistent with other sub-genomic analyses in maize, small RNA mapping showed that many small RNAs match TEs and that most TEs match small RNAs. These results, performed on ∼1% of the maize genome, demonstrate the feasibility of refining the B73 RefGen_v1 genome assembly by incorporating optical map, high-resolution genetic map, and comparative genomic data sets. Such improvements, along with those of gene and repeat annotation, will serve to promote future functional genomic and phylogenomic research in maize and other grasses