The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law

Control of flowering time and spike development in cereals: the earliness per se Eps-1 region in wheat, rice, and Brachypodium

The earliness per se gene Eps-Am1 from diploid wheat Triticum monococcum affects heading time, spike development, and spikelet number. In this study, the Eps1 orthologous regions from rice, Aegilops tauschii, and Brachypodium distachyon were compared as part of current efforts to clone this gene. A single Brachypodium BAC clone spanned the Eps-Am1 region, but a gap was detected in the A. tauschii physical map. Sequencing of the Brachypodium and A. tauschii BAC clones revealed three genes shared by the three species, which showed higher identity between wheat and Brachypodium than between them and rice. However, most of the structural changes were detected in the wheat lineage. These included an inversion encompassing the wg241-VatpC region and the presence of six unique genes. In contrast, only one unique gene (and one pseudogene) was found in Brachypodium and none in rice. Three genes were present in both Brachypodium and wheat but were absent in rice. Two of these genes, Mot1 and FtsH4, were completely linked to the earliness per se phenotype in the T. monococcum high-density genetic map and are candidates for Eps-Am1. Both genes were expressed in apices and developing spikes, as expected for Eps-Am1 candidates. The predicted MOT1 protein showed amino acid differences between the parental T. monococcum lines, but its effect is difficult to predict. Future steps to clone the Eps-Am1 gene include the generation of mot1 and ftsh4 mutants and the completion of the T. monococcum physical map to test for the presence of additional candidate genes

Duplication of a well-conserved homeodomain-leucine zipper transcription factor gene in barley generates a copy with more specific functions

Three spikelets are formed at each rachis node of the cultivated barley (Hordeum vulgare ssp. vulgare) spike. In two-rowed barley, the central one is fertile and the two lateral ones are sterile, whereas in the six-rowed type, all three are fertile. This characteristic is determined by the allelic constitution at the six-rowed spike 1 (vrs1) locus on the long arm of chromosome 2H, with the recessive allele (vrs1) being responsible for the six-rowed phenotype. The Vrs1 (HvHox1) gene encodes a homeodomain-leucine zipper (HD-Zip) transcription factor. Here, we show that the Vrs1 gene evolved in the Poaceae via a duplication, with a second copy of the gene, HvHox2, present on the short arm of chromosome 2H. Micro-collinearity and polypeptide sequences were both well conserved between HvHox2 and its Poaceae orthologs, but Vrs1 is unique to the barley tribe. The Vrs1 gene product lacks a motif which is conserved among the HvHox2 orthologs. A phylogenetic analysis demonstrated that Vrs1 and HvHox2 must have diverged after the separation of Brachypodium distachyon from the Pooideae and suggests that Vrs1 arose following the duplication of HvHox2, and acquired its new function during the evolution of the barley tribe. HvHox2 was expressed in all organs examined but Vrs1 was predominantly expressed in immature inflorescence

Patterns of Sequence Divergence and Evolution of the S1 Orthologous Regions between Asian and African Cultivated Rice Species

A strong postzygotic reproductive barrier separates the recently diverged Asian and African cultivated rice species, Oryza sativa and O. glaberrima. Recently a model of genetic incompatibilities between three adjacent loci: S1A, S1 and S1B (called together the S1 regions) interacting epistatically, was postulated to cause the allelic elimination of female gametes in interspecific hybrids. Two candidate factors for the S1 locus (including a putative F-box gene) were proposed, but candidates for S1A and S1B remained undetermined. Here, to better understand the basis of the evolution of regions involved in reproductive isolation, we studied the genic and structural changes accumulated in the S1 regions between orthologous sequences. First, we established an 813 kb genomic sequence in O. glaberrima, covering completely the S1A, S1 and the majority of the S1B regions, and compared it with the orthologous regions of O. sativa. An overall strong structural conservation was observed, with the exception of three isolated regions of disturbed collinearity: (1) a local invasion of transposable elements around a putative F-box gene within S1, (2) the multiple duplication and subsequent divergence of the same F-box gene within S1A, (3) an interspecific chromosomal inversion in S1B, which restricts recombination in our O. sativa×O. glaberrima crosses. Beside these few structural variations, a uniform conservative pattern of coding sequence divergence was found all along the S1 regions. Hence, the S1 regions have undergone no drastic variation in their recent divergence and evolution between O. sativa and O. glaberrima, suggesting that a small accumulation of genic changes, following a Bateson-Dobzhansky-Muller (BDM) model, might be involved in the establishment of the sterility barrier. In this context, genetic incompatibilities involving the duplicated F-box genes as putative candidates, and a possible strengthening step involving the chromosomal inversion might participate to the reproductive barrier between Asian and African rice species

An atlas of over 90.000 conserved noncoding sequences provides insight into crucifer regulatory regions

Despite the central importance of noncoding DNA to gene regulation and evolution, understanding of the extent of selection on plant noncoding DNA remains limited compared to that of other organisms. Here we report sequencing of genomes from three Brassicaceae species (Leavenworthia alabamica, Sisymbrium irio and Aethionema arabicum) and their joint analysis with six previously sequenced crucifer genomes. Conservation across orthologous bases suggests that at least 17% of the Arabidopsis thaliana genome is under selection, with nearly one-quarter of the sequence under selection lying outside of coding regions. Much of this sequence can be localized to approximately 90,000 conserved noncoding sequences (CNSs) that show evidence of transcriptional and post-transcriptional regulation. Population genomics analyses of two crucifer species, A. thaliana and Capsella grandiflora, confirm that most of the identified CNSs are evolving under medium to strong purifying selection. Overall, these CNSs highlight both similarities and several key differences between the regulatory DNA of plants and other species