The Primary Pretenders

We call a composite number q such that there exists a positive integer b with b^p == b (mod q) a prime pretender to base b. The least prime pretender to base b is the primary pretender q_b. It is shown that there are only 132 distinct primary pretenders, and that q_b is a periodic function of b whose period is the 122-digit number 19568584333460072587245340037736278982017213829337604336734362- 294738647777395483196097971852999259921329236506842360439300.Comment: 7 page

Making irrigation investments pay

Cover title.Includes bibliographical references

Rapid and Inexpensive Whole-Genome Genotyping-by-Sequencing for Crossover Localization and Fine-Scale Genetic Mapping

The reshuffling of existing genetic variation during meiosis is important both during evolution and in breeding. The reassortment of genetic variants relies on the formation of crossovers (COs) between homologous chromosomes. The pattern of genome-wide CO distributions can be rapidly and precisely established by the short-read sequencing of individuals from F(2) populations, which in turn are useful for quantitative trait locus (QTL) mapping. Although sequencing costs have decreased precipitously in recent years, the costs of library preparation for hundreds of individuals have remained high. To enable rapid and inexpensive CO detection and QTL mapping using low-coverage whole-genome sequencing of large mapping populations, we have developed a new method for library preparation along with Trained Individual GenomE Reconstruction, a probabilistic method for genotype and CO predictions for recombinant individuals. In an example case with hundreds of F(2) individuals from two Arabidopsis thaliana accessions, we resolved most CO breakpoints to within 2 kb and reduced a major flowering time QTL to a 9-kb interval. In addition, an extended region of unusually low recombination revealed a 1.8-Mb inversion polymorphism on the long arm of chromosome 4. We observed no significant differences in the frequency and distribution of COs between F(2) individuals with and without a functional copy of the DNA helicase gene RECQ4A. In summary, we present a new, cost-efficient method for large-scale, high-precision genotyping-by-sequencing

The Striking Flower-in-Flower Phenotype of Arabidopsis thaliana Nossen (No-0) is Caused by a Novel LEAFY Allele

The transition to reproduction is a crucial step in the life cycle of any organism. In Arabidopsis thaliana the establishment of reproductive growth can be divided into two phases: Firstly, cauline leaves with axillary meristems are formed and internode elongation begins. Secondly, lateral meristems develop into flowers with defined organs. Floral shoots are usually determinate and suppress the development of lateral shoots. Here, we describe a transposon insertion mutant in the Nossen accession with defects in floral development and growth. Most strikingly is the outgrowth of stems from the axillary bracts of the primary flower carrying secondary flowers. Therefore, we named this mutant flower-in-flower (fif). However, the transposon insertion in the annotated gene is not the cause for the fif phenotype. By means of classical and genome sequencing-based mapping, the mutation responsible for the fif phenotype was found to be in the LEAFY gene. The mutation, a G-to-A exchange in the second exon of LEAFY, creates a novel lfy allele and results in a cysteine-to-tyrosine exchange in the α1-helix of LEAFY’s DNA-binding domain. This exchange abolishes target DNA-binding, whereas subcellular localization and homomerization are not affected. To explain the strong fif phenotype against these molecular findings, several hypotheses are discussed

Chromosome-scale and haplotype-resolved genome assembly of a tetraploid potato cultivar

Potato is the most widely produced tuber crop worldwide. However, reconstructing the four haplotypes of its autotetraploid genome remained an unsolved challenge. Here, we report the 3.1 Gb haplotype-resolved (at 99.6% precision), chromosome-scale assembly of the potato cultivar ‘Otava’ based on high-quality long reads, single-cell sequencing of 717 pollen genomes and Hi-C data. Unexpectedly, ~50% of the genome was identical-by-descent due to recent inbreeding, which was contrasted by highly abundant structural rearrangements involving ~20% of the genome. Among 38,214 genes, only 54% were present in all four haplotypes with an average of 3.2 copies per gene. Taking the leaf transcriptome as an example, 11% of the genes were differently expressed in at least one haplotype, where 25% of them were likely regulated through allele-specific DNA methylation. Our work sheds light on the recent breeding history of potato, the functional organization of its tetraploid genome and has the potential to strengthen the future of genomics-assisted breeding