Toric Border Bases

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining "toric" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals

Stable normal forms for polynomial system solving

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper

Border basis representation of a general quotient algebra

International audienceIn this paper, we generalized the construction of border bases to non-zero dimensional ideals for normal forms compatible with the degree, tackling the remaining obstacle for a general application of border basis methods. First, we give conditions to have a border basis up to a given degree. Next, we describe a new stopping criteria to determine when the reduction with respect to the leading terms is a normal form. This test based on the persistence and regularity theorems of Gotzmann yields a new algorithm for computing a border basis of any ideal, which proceeds incrementally degree by degree until its regularity. We detail it, prove its correctness, present its implementation and report some experimentations which illustrate its practical good behavior

A single transgene locus triggers both transcriptional and post-transcriptional silencing through double-stranded RNA production

Silencing of a target locus by an unlinked silencing locus can result from transcription inhibition (transcriptional gene silencing; TGS) or mRNA degradation (post-transcriptional gene silencing; PTGS), owing to the production of double-stranded RNA (dsRNA) corresponding to promoter or transcribed sequences, respectively. The involvement of distinct cellular components in each process suggests that dsRNA-induced TGS and PTGS likely result from the diversification of an ancient common mechanism. However, a strict comparison of TGS and PTGS has been difficult to achieve because it generally relies on the analysis of distinct silencing loci. We describe a single transgene locus that triggers both TGS and PTGS, owing to the production of dsRNA corresponding to promoter and transcribed sequences of different target genes. We describe mutants and epigenetic variants derived from this locus and propose a model for the production of dsRNA. Also, we show that PTGS, but not TGS, is graft-transmissible, which together with the sensitivity of PTGS, but not TGS, to RNA viruses that replicate in the cytoplasm, suggest that the nuclear compartmentalization of TGS is responsible for cell-autonomy. In contrast, we contribute local and systemic trafficking of silencing signals and sensitivity to viruses to the cytoplasmic steps of PTGS and to amplification steps that require high levels of target mRNAs

Circular Cylinders by Four or Five Points in Space

International audienceWe are interested in computing effectively cylinders through 5 points, and in other problems involved in metrology. In particular, we consider the cylinders through 4 points with a fix radius and with extremal radius. For these different problems, we give bounds on the number of solutions and exemples show that these bounds are optimal. Finally, we describe two algebraic methods which can be used here to solve efficiently these problems and some experimentation results

Molecular integration of casanova in the Nodal signalling pathway controlling endoderm formation

International audienceEndoderm originates from a large endomesodermal field requiring Nodal signalling. The mechanisms that ensure segregation of endoderm from mesoderm are not fully understood. We first show that the timing and dose of Nodal activation are crucial for endoderm formation and the endoderm versus mesoderm fate choice, because sustained Nodal signalling is required to ensure endoderm formation but transient signalling is sufficient for mesoderm formation. In zebrafish, downstream of Nodal signals, three genes encoding transcription factors (faust, bonnie and clyde and the recently identified gene casanova) are required for endoderm formation and differentiation. However their positions within the pathway are not completely established. In the present work, we show that casanova is the earliest specification marker for endodermal cells and that its expression requires bonnie and clyde. Furthermore, we have analysed the molecular activities of casanova on endoderm formation and found that it can induce endodermal markers and repress mesodermal markers during gastrulation, as well as change the fate of marginal blastomeres to endoderm. Overexpression of casanova also restores endoderm markers in the absence of Nodal signalling. In addition, casanova efficiently restores later endodermal differentiation in these mutants, but this process requires, in addition, a partial activation of Nodal signalling

Symbolic Methods for Solving Algebraic Systems of Equations and Applications for Testing the Structural Stability

International audienceIn this work, we provide an overview of the classical symbolic techniques for solving algebraic systems of equations and show the interest of such techniques in the study of some problems in dynamical system theory, namely testing the structural stability of multidimensional systems

Characterization of Sleep in Zebrafish and Insomnia in Hypocretin Receptor Mutants

Sleep is a fundamental biological process conserved across the animal kingdom. The study of how sleep regulatory networks are conserved is needed to better understand sleep across evolution. We present a detailed description of a sleep state in adult zebrafish characterized by reversible periods of immobility, increased arousal threshold, and place preference. Rest deprivation using gentle electrical stimulation is followed by a sleep rebound, indicating homeostatic regulation. In contrast to mammals and similarly to birds, light suppresses sleep in zebrafish, with no evidence for a sleep rebound. We also identify a null mutation in the sole receptor for the wake-promoting neuropeptide hypocretin (orexin) in zebrafish. Fish lacking this receptor demonstrate short and fragmented sleep in the dark, in striking contrast to the excessive sleepiness and cataplexy of narcolepsy in mammals. Consistent with this observation, we find that the hypocretin receptor does not colocalize with known major wake-promoting monoaminergic and cholinergic cell groups in the zebrafish. Instead, it colocalizes with large populations of GABAergic neurons, including a subpopulation of Adra2a-positive GABAergic cells in the anterior hypothalamic area, neurons that could assume a sleep modulatory role. Our study validates the use of zebrafish for the study of sleep and indicates molecular diversity in sleep regulatory networks across vertebrates

Rasl11b Knock Down in Zebrafish Suppresses One-Eyed-Pinhead Mutant Phenotype

The EGF-CFC factor Oep/Cripto1/Frl1 has been implicated in embryogenesis and several human cancers. During vertebrate development, Oep/Cripto1/Frl1 has been shown to act as an essential coreceptor in the TGFβ/Nodal pathway, which is crucial for germ layer formation. Although studies in cell cultures suggest that Oep/Cripto1/Frl1 is also implicated in other pathways, in vivo it is solely regarded as a Nodal coreceptor. We have found that Rasl11b, a small GTPase belonging to a Ras subfamily of putative tumor suppressor genes, modulates Oep function in zebrafish independently of the Nodal pathway. rasl11b down regulation partially rescues endodermal and prechordal plate defects of zygotic oep−/− mutants (Zoep). Rasl11b inhibitory action was only observed in oep-deficient backgrounds, suggesting that normal oep expression prevents Rasl11b function. Surprisingly, rasl11b down regulation does not rescue mesendodermal defects in other Nodal pathway mutants, nor does it influence the phosphorylation state of the downstream effector Smad2. Thus, Rasl11b modifies the effect of Oep on mesendoderm development independently of the main known Oep output: the Nodal signaling pathway. This data suggests a new branch of Oep signaling that has implications for germ layer development, as well as for studies of Oep/Frl1/Cripto1 dysfunction, such as that found in tumors

Regulation of High-Temperature Stress Response by Small RNAs

Temperature extremes constitute one of the most common environmental stresses that adversely affect the growth and development of plants. Transcriptional regulation of temperature stress responses, particularly involving protein-coding gene networks, has been intensively studied in recent years. High-throughput sequencing technologies enabled the detection of a great number of small RNAs that have been found to change during and following temperature stress. The precise molecular action of some of these has been elucidated in detail. In the present chapter, we summarize the current understanding of small RNA-mediated modulation of high- temperature stress-regulatory pathways including basal stress responses, acclimation, and thermo-memory. We gather evidence that suggests that small RNA network changes, involving multiple upregulated and downregulated small RNAs, balance the trade-off between growth/development and stress responses, in order to ensure successful adaptation. We highlight specific characteristics of small RNA-based tem- perature stress regulation in crop plants. Finally, we explore the perspectives of the use of small RNAs in breeding to improve stress tolerance, which may be relevant for agriculture in the near future