A unique pair of triangles

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic abelian integrals.Comment: 5 pages, to appear in Journal of Number Theory, Some modifications are added to the article published onlin

The persimmon genome reveals clues to the evolution of a lineage-specific sex determination system in plants

Most angiosperms bear hermaphroditic flowers, but a few species have evolved outcrossing strategies, such as dioecy, the presence of separate male and female individuals. We previously investigated the mechanisms underlying dioecy in diploid persimmon (D. lotus) and found that male flowers are specified by repression of the autosomal gene MeGI by its paralog, the Y-encoded pseudo-gene OGI. This mechanism is thought to be lineage-specific, but its evolutionary path remains unknown. Here, we developed a full draft of the diploid persimmon genome (D. lotus), which revealed a lineage-specific whole-genome duplication event and provided information on the architecture of the Y chromosome. We also identified three paralogs, MeGI, OGI and newly identified Sister of MeGI (SiMeGI). Evolutionary analysis suggested that MeGI underwent adaptive evolution after the whole-genome duplication event. Transformation of tobacco plants with MeGI and SiMeGI revealed that MeGI specifically acquired a new function as a repressor of male organ development, while SiMeGI presumably maintained the original function. Later, a segmental duplication event spawned MeGI's regulator OGI on the Y-chromosome, completing the path leading to dioecy, and probably initiating the formation of the Y-chromosome. These findings exemplify how duplication events can provide flexible genetic material available to help respond to varying environments and provide interesting parallels for our understanding of the mechanisms underlying the transition into dieocy in plants. Author summary Plant sexuality has fascinated scientists for decades. Most plants can self-reproduce but not all. For example, a small subset of species have evolved a system called dioecy, with separate male and female individuals. Dioecy has evolved multiple times independently and, while we do not understand the molecular mechanisms underlying dioecy in many of these species yet, a picture is starting to emerge with recent progress in several dioecious species. Here, we focused on the evolutionary events leading to dioecy in persimmon. Our previous work had identified a pair of genes regulating sex in this species, called OGI and MeGI. We drafted the whole genome sequence of diploid persimmon to investigate their evolutionary history. We discovered a lineage-specific whole-genome duplication event, and observed that MeGI underwent adaptive evolution after this event. Transgenic analyses validated that MeGI newly acquired a male-suppressor function, while the other copy of this gene, SiMeGI, did not. The regulator of MeGI, OGI, resulted from a second smaller-scale segmental duplication event, finalizing the system. This study sheds light on the role of duplication as a mechanism that promote flexible genes functions, and how it can affect important biological functions, such as the establishment of a new sexual system

Genome-wide Analysis of Chlamydophila pneumoniae Gene Expression at the Late Stage of Infection

Chlamydophila pneumoniae, an obligate intracellular eubacterium, changes its form from a vegetative reticulate body into an infectious elementary body during the late stage of its infection cycle. Comprehension of the molecular events in the morphological change is important to understand the switching mechanism between acute and chronic infection, which is deemed to relate to the pathogenesis of atherosclerosis. Herein, we have attempted to screen genes expressed in the late stage with a genome-wide DNA microarray, resulting in nomination of 17 genes as the late-stage genes. Fourteen of the 17 genes and six other genes predicted as late-stage genes were confirmed to be up-regulated in the late stage with a quantitative reverse transcriptase–polymerase chain reaction. These 20 late-stage genes were classified into two groups by clustering analysis: ‘drastically induced’ and ‘moderately induced’ genes. Out of eight drastically induced genes, four contain σ28 promoter-like sequences and the other four contain an upstream common sequence. It suggests that besides σ28, there are certain up-regulatory mechanisms at the late stage, which may be involved in the chlamydial morphological change and thus pathogenesis

X chromosome-wide analyses of genomic DNA methylation states and gene expression in male and female neutrophils

The DNA methylation status of human X chromosomes from male and female neutrophils was identified by high-throughput sequencing of HpaII and MspI digested fragments. In the intergenic and intragenic regions on the X chromosome, the sites outside CpG islands were heavily hypermethylated to the same degree in both genders. Nearly half of X chromosome promoters were either hypomethylated or hypermethylated in both females and males. Nearly one third of X chromosome promoters were a mixture of hypomethylated and heterogeneously methylated sites in females and were hypomethylated in males. Thus, a large fraction of genes that are silenced on the inactive X chromosome are hypomethylated in their promoter regions. These genes frequently belong to the evolutionarily younger strata of the X chromosome. The promoters that were hypomethylated at more than two sites contained most of the genes that escaped silencing on the inactive X chromosome. The overall levels of expression of X-linked genes were indistinguishable in females and males, regardless of the methylation state of the inactive X chromosome. Thus, in addition to DNA methylation, other factors are involved in the fine tuning of gene dosage compensation in neutrophils

The Lifestyle of the Segmented Filamentous Bacterium: A Non-Culturable Gut-Associated Immunostimulating Microbe Inferred by Whole-Genome Sequencing

Numerous microbes inhabit the mammalian intestinal track and strongly impact host physiology; however, our understanding of this ecosystem remains limited owing to the high complexity of the microbial community and the presence of numerous non-culturable microbes. Segmented filamentous bacteria (SFBs), which are clostridia-related Gram-positive bacteria, are among such non-culturable populations and are well known for their unique morphology and tight attachment to intestinal epithelial cells. Recent studies have revealed that SFBs play crucial roles in the post-natal maturation of gut immune function, especially the induction of Th17 lymphocytes. Here, we report the complete genome sequence of mouse SFBs. The genome, which comprises a single circular chromosome of 1 620 005 bp, lacks genes for the biosynthesis of almost all amino acids, vitamins/cofactors and nucleotides, but contains a full set of genes for sporulation/germination and, unexpectedly, for chemotaxis/flagella-based motility. These findings suggest a triphasic lifestyle of the SFB, which comprises two types of vegetative (swimming and epicellular parasitic) phases and a dormant (spore) phase. Furthermore, SFBs encode four types of flagellin, three of which are recognized by Toll-like receptor 5 and could elicit the innate immune response. Our results reveal the non-culturability, lifestyle and immunostimulation mechanisms of SFBs and provide a genetic basis for the future development of the SFB cultivation and gene-manipulation techniques

Comparative Genetic Mapping and Discovery of Linkage Disequilibrium Across Linkage Groups in White Clover (Trifolium repens L.)

White clover (Trifolium repens L.) is an allotetraploid species (2n = 4X = 32) that is widely distributed in temperate regions and cultivated as a forage legume. In this study, we developed expressed sequence tag (EST)–derived simple sequence repeat (SSR) markers, constructed linkage maps, and performed comparative mapping with other legume species. A total of 7982 ESTs that could be assembled into 5400 contigs and 2582 singletons were generated. Using the EST sequences that were obtained, 1973 primer pairs to amplify EST-derived SSR markers were designed and used for linkage analysis of 188 F1 progenies, which were generated by a cross between two Japanese plants, ‘273-7’ and ‘T17-349,’ with previously published SSR markers. An integrated linkage map was constructed by combining parental-specific maps, which consisted of 1743 SSR loci on 16 homeologous linkage groups with a total length of 2511 cM. The primer sequences of the developed EST-SSR markers and their map positions are available on http://clovergarden.jp/. Linkage disequilibrium (LD) was observed on 9 of 16 linkage groups of a parental-specific map. The genome structures were compared among white clover, red clover (T. pratense L.), Medicago truncatula, and Lotus japonicus. Macrosynteny was observed across the four legume species. Surprisingly, the comparative genome structure between white clover and M. truncatula had a higher degree of conservation than that of the two clover species