Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization

We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure

WeederH: an algorithm for finding conserved regulatory motifs and regions in homologous sequences

BACKGROUND: This work addresses the problem of detecting conserved transcription factor binding sites and in general regulatory regions through the analysis of sequences from homologous genes, an approach that is becoming more and more widely used given the ever increasing amount of genomic data available. RESULTS: We present an algorithm that identifies conserved transcription factor binding sites in a given sequence by comparing it to one or more homologs, adapting a framework we previously introduced for the discovery of sites in sequences from co-regulated genes. Differently from the most commonly used methods, the approach we present does not need or compute an alignment of the sequences investigated, nor resorts to descriptors of the binding specificity of known transcription factors. The main novel idea we introduce is a relative measure of conservation, assuming that true functional elements should present a higher level of conservation with respect to the rest of the sequence surrounding them. We present tests where we applied the algorithm to the identification of conserved annotated sites in homologous promoters, as well as in distal regions like enhancers. CONCLUSION: Results of the tests show how the algorithm can provide fast and reliable predictions of conserved transcription factor binding sites regulating the transcription of a gene, with better performances than other available methods for the same task. We also show examples on how the algorithm can be successfully employed when promoter annotations of the genes investigated are missing, or when regulatory sites and regions are located far away from the genes

Differential Coupling of Self-Renewal Signaling Pathways in Murine Induced Pluripotent Stem Cells

The ability to reprogram somatic cells to induced pluripotent stem cells (iPSCs), exhibiting properties similar to those of embryonic stem cells (ESCs), has attracted much attention, with many studies focused on improving efficiency of derivation and unraveling the mechanisms of reprogramming. Despite this widespread interest, our knowledge of the molecular signaling pathways that are active in iPSCs and that play a role in controlling their fate have not been studied in detail. To address this shortfall, we have characterized the influence of different signals on the behavior of a model mouse iPSC line. We demonstrate significant responses of this iPSC line to the presence of serum, which leads to profoundly enhanced proliferation and, depending on the medium used, a reduction in the capacity of the iPSCs to self-renew. Surprisingly, this iPSC line was less sensitive to withdrawal of LIF compared to ESCs, exemplified by maintenance of expression of a Nanog-GFP reporter and enhanced self-renewal in the absence of LIF. While inhibition of phosphoinositide-3 kinase (PI3K) signaling decreased iPSC self-renewal, inhibition of Gsk-3 promoted it, even in the absence of LIF. High passages of this iPSC line displayed altered characteristics, including genetic instability and a reduced ability to self-renew. However, this second feature could be restored upon inhibition of Gsk-3. Collectively, our data suggest modulation of Gsk-3 activity plays a key role in the control of iPSC fate. We propose that more careful consideration should be given to characterization of the molecular pathways that control the fate of different iPSC lines, since perturbations from those observed in naïve pluripotent ESCs could render iPSCs and their derivatives susceptible to aberrant and potentially undesirable behaviors

Inactivation of a Single Copy of Crebbp Selectively Alters Pre-mRNA Processing in Mouse Hematopoietic Stem Cells

Global expression analysis of fetal liver hematopoietic stem cells (FL HSCs) revealed the presence of unspliced pre-mRNA for a number of genes in normal FL HSCs. In a subset of these genes, Crebbp+/− FL HSCs had less unprocessed pre-mRNA without a corresponding reduction in total mRNA levels. Among the genes thus identified were the key regulators of HSC function Itga4, Msi2 and Tcf4. A similar but much weaker effect was apparent in Ep300+/− FL HSCs, indicating that, in this context as in others, the two paralogs are not interchangeable. As a group, the down-regulated intronic probe sets could discriminate adult HSCs from more mature cell types, suggesting that the underlying mechanism is regulated with differentiation stage and is active in both fetal and adult hematopoiesis. Consistent with increased myelopoiesis in Crebbp hemizygous mice, targeted reduction of CREBBP abundance by shRNA in the multipotent EML cell line triggered spontaneous myeloid differentiation in the absence of the normally required inductive signals. In addition, differences in protein levels between phenotypically distinct EML subpopulations were better predicted by taking into account not only the total mRNA signal but also the amount of unspliced message present. CREBBP thus appears to selectively influence the timing and degree of pre-mRNA processing of genes essential for HSC regulation and thereby has the potential to alter subsequent cell fate decisions in HSCs

Inequalities for products of polynomials

Abstract. We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment [−1, 1]. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane. 1. The problem and its history Let E be a compact set in the complex plane C. For a function f: E → C define the uniform (sup) norm as follows: �f�E = sup |f(z)|. z∈E Clearly �f1f2 � E ≤ �f1 � E �f2 � E, but this inequality is not reversible, in general, not even with a constant factor in front of the right hand side. Indeed, �f1 � E �f2 � E ≤ C �f1f2 � E does not hold for functions with disjoint supports in E, for example. However, the situation is quite different for algebraic polyno-mials {pk(z)} m k=1 and their product p(z): = �m k=1 pk(z). Polynomial inequalities of the form m� (1.1) �pk�E ≤ C�p�E, k=1 exist and are readily available. One of the first results in this direction is due to Kneser [19], for E = [−1, 1] and m = 2 (see also Aumann [1]), who proved that (1.2) �p1�[−1,1]�p2�[−1,1] ≤ Kℓ,n�p1p2�[−1,1], deg p1 = ℓ, deg p2 = n − ℓ, 2000 Mathematics Subject Classification. Primary 30C10; Secondary 30C85, 31A15. Key words and phrases. Polynomials, products, factors, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points. Research of I.P. was partially supported by the National Security Agency (grant H98230-06-1-0055), and by the Alexander von Humboldt Foundation. S.R. acknowledges partial support from the German-Israeli Foundation (grant G-809-234.6/2003), from FONDECYT (grants 1040366 and 7040069) and from DGIP-UTFSM (grant 240104).