Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$\mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x];$$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ stands for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519

Rank one discrete valuations of power series fields

In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in k\lcor\X\rcor is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the Zorn's lemma or to know explicitly a section $\sigma$ of the natural homomorphism R_v\to\d between the ring and the residue field of the valuation $v$. However, as a consequence of this construction, in section 7, we prove that k((\X)) can be embedded into a field L((\Y)), where $L$ is an algebraic extension of $k$ and the {\em ``extended valuation'' is as close as possible to the usual order function}

Requirements for translation re-initiation in Escherichia coli: roles of initiator tRNA and initiation factors IF2 and IF3

Despite its importance in post-transcriptional regulation of polycistronic operons in Escherichia coli, little is known about the mechanism of translation re-initiation, which occurs when the same ribosome used to translate an upstream open reading frame (ORF) also translates a downstream ORF. To investigate translation re-initiation in Escherichia coli, we constructed a di-cistronic reporter in which a firefly luciferase gene was linked to a chloramphenicol acetyltransferase gene using a segment of the translationally coupled geneV–geneVII intercistronic region from M13 phage. With this reporter and mutant initiator tRNAs, we show that two of the unique properties of E. coli initiator tRNA – formylation of the amino acid attached to the tRNA and binding of the tRNA to the ribosomal P-site – are as important for re-initiation as for de novo initiation. Overexpression of IF2 or increasing the affinity of mutant initiator tRNA for IF2 enhanced re-initiation efficiency, suggesting that IF2 is required for efficient re-initiation. In contrast, overexpression of IF3 led to a marked decrease in re-initiation efficiency, suggesting that a 30S ribosome and not a 70S ribosome is used for translation re-initiation. Strikingly, overexpression of IF3 also blocked E. coli from acting as a host for propagation of M13 phage

Network analysis of the transcriptional pattern of young and old cells of Escherichia coli during lag phase

Background: The aging process of bacteria in stationary phase is halted if cells are subcultured and enter lag phase and it is then followed by cellular division. Network science has been applied to analyse the transcriptional response, during lag phase, of bacterial cells starved previously in stationary phase for 1 day (young cells) and 16 days (old cells). Results: A genome scale network was constructed for E. coli K-12 by connecting genes with operons, transcription and sigma factors, metabolic pathways and cell functional categories. Most of the transcriptional changes were detected immediately upon entering lag phase and were maintained throughout this period. The lag period was longer for older cells and the analysis of the transcriptome revealed different intracellular activity in young and old cells. The number of genes differentially expressed was smaller in old cells (186) than in young cells (467). Relatively, few genes (62) were up- or down-regulated in both cultures. Transcription of genes related to osmotolerance, acid resistance, oxidative stress and adaptation to other stresses was down-regulated in both young and old cells. Regarding carbohydrate metabolism, genes related to the citrate cycle were up-regulated in young cells while old cells up-regulated the Entner Doudoroff and gluconate pathways and down-regulated the pentose phosphate pathway. In both old and young cells, anaerobic respiration and fermentation pathways were down-regulated, but only young cells up-regulated aerobic respiration while there was no evidence of aerobic respiration in old cells.Numerous genes related to DNA maintenance and replication, translation, ribosomal biosynthesis and RNA processing as well as biosynthesis of the cell envelope and flagellum and several components of the chemotaxis signal transduction complex were up-regulated only in young cells. The genes for several transport proteins for iron compounds were up-regulated in both young and old cells. Numerous genes encoding transporters for carbohydrates and organic alcohols and acids were down-regulated in old cells only. Conclusion: Network analysis revealed very different transcriptional activities during the lag period in old and young cells. Rejuvenation seems to take place during exponential growth by replicative dilution of old cellular components

Mechanism of translational coupling in the nifLA operon of Klebsiella pneumoniae.

The nifLA operon of Klebsiella pneumoniae encodes the sensor-activator pair involved in the regulation of other nif genes. Balanced synthesis of both proteins, which is required for correct regulation, is achieved by coupling translation of nifA to that of nifL. The mechanism of translational coupling at the nifLA operon was analysed using a specialized ribosome system, and the effect of substituting the natural Shine-Dalgarno of nifL or nifA for specialized Shine-Dalgarno sequences was determined. Our results indicate that translational coupling occurs in this operon by a reinitiation mechanism. Additionally, reinitiation at the nifA can happen even in the absence of good Shine-Dalgarno recognition by the reinitiating ribosome, although its efficiency is lower. The effect of a putative translational enhancer sequence (downstream box) on translational coupling efficiency was also determined. Mutations that reduce the homology of the putative downstream box to the consensus had only a minor effect on nifA translation by wild-type ribosomes. However, they had a significant effect on nifA translation by specialized ribosomes, suggesting that recognition of the downstream box may compensate inefficient ribosomal interactions with the Shine-Dalgarno sequence

Mechanism of coordinated synthesis of the antagonistic regulatory proteins NifL and NifA of Klebsiella pneumoniae.

The nifLA operon of Klebsiella pneumoniae codes for the two antagonistic regulatory proteins which control expression of all other nitrogen fixation genes. NifA is a transcriptional activator, and NifL inhibits NifA. The importance of a correct NifL-NifA stoichiometry for efficient regulation of nitrogen fixation genes has been investigated by constructing a strain with an altered nifL-nifA gene dosage ratio, resulting from the integration of an extra copy of nifA. Results showed that a balanced synthesis of both gene products is essential for correct regulation. Effects of mutations provoking translation termination of nifL upstream or downstream of its natural stop codon, combined with overproduction of both proteins when the genes are transcribed and translated from signals of the phi10 gene of the phage T7, showed that, in addition to the previously reported transcriptional polarity, there is translational coupling between nifL and nifA. In spite of the apparently efficient ribosome binding site of nifA, its rate of independent translation is very low. This is due to a secondary structure masking the Shine-Dalgarno sequence of nifA, which could be melted by ribosomes translating nifL. Mutational analysis confirmed the functional significance of the secondary structure in preventing independent translation of nifA. Translational coupling between the two cistrons is proposed as an efficient mechanism to prevent production of an excess of NifA, which would affect the normal regulation of nitrogen fixation genes